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Changing inner text value of tab through javascript

I'm learning Javascript right now, and attempting to change the text title of a particular tab. It's actually part of a larger Shiny dashboard project, but I want to add some custom functionality to a few tabs. Below are the tabs in question:
Simple enough. I first access my tabs in my Javascript file:
var tabScrub2 = $(document).find('[data-value="scrubTab2"]');
console.log(tabScrub2);
When I use Firefox's developer console, I see that the tab is an object:
Moreover, it looks like I need to change the innerText property of 0, whatever this is, since that corresponds to the title of my tab (the innerText of 1 corresponds to the text inside scrubTab2). However, I'm not familiar with the actual object type being returned here:
Simply put, how the heck do I access and manipulate properties from this? And am I actually accessing an array? When I type in
var scrub2 = tabScrub2["1"];
console.log(scrub2);
I get an HTML element. I'm seen the a element in CSS and jQuery, but am not super familiar with how to manipulate its properties programmatically? How do I go about accessing and manipulating the innerText properties of this via Javascript? For instance, how would I hide scrubTab2, or change its title to something else?

The first object you're seeing is jQuery's wrapper around the real DOM elements. It's not an actual array, but it does contain all of the elements that matched your query under zero-indexed properties (e.g. "0" and "1") which allows you to access to them via an array-like API (e.g. tabScrub[1]).
Your method of grabbing a node using tabScrub2["1"] is correct (see this question in the jQuery FAQ). It's more likely to see that done with a numeric key though (i.e. tabScrub[1]) because that matches the way you would access an element in a normal array.
As far as manipulating properties of the DOM node, the DOM's API is notoriously inconsistent and quirky (hence the need for things like jQuery in the first place). However, for your use case you can just assign a string to the innerText property directly (e.g. tagScrub2[1].innerText = "Tab title"). MDN is a great resource if you're looking for reference material on other parts of the DOM.
A side note: if you're looking for a specific element you should use a query that will only match that element. It's generally a bad sign if you're grabbing extra elements and then accessing the element you want at a key other than 0. If you're doing this then your code depends on other (potentially unrelated) nodes in the DOM existing before your node, and if/when you change those nodes your original code will break.

Just use jQuery eq method to get the relevant object index from the array.
For an example
//Query and get first element.
var tabScrub2 = $(document).find('[data-value="scrubTab2"]:eq(0)');
//Hide
tabScrub2.hide();
//Change title
tabScrub2.attr("title", "New Title Text");
Lean more about jQuery eq here.
https://api.jquery.com/eq/
Since you use jquery selectors tabScrub2[0] returns the native DOM element instead of another jQuery object. Therefore the hide function won't work in that object since the native DOM element doesn't implement such type of functionality for an element. That's why you have to use jQuery pseudo selector as above. Because hide will only work with a jQuery object.

How to tell the type of a JavaScript and/or jQueryobject

This question pertains as much to the ECMAScript language implementation we know as JavaScript as it does to jQuery and the developer tools availble in most popular browsers.
When you execute a statement like so:
var theElement = $('#theId').closest();
what is the type of theElement?
I assume that in a jQuery situation like above, many jQuery methods including the one above actually return the jQuery object itself, which packages the stuff you actually want to get to. This, so that it may maintain a fluent API and let you join method calls in a single statement like so:
$('#selector').foo().bar().gar().har();
However, in the case of jQuery then, how do you determine what the real underlying type is? For example, if the element returned was a table row with the Id tableRowNumber25, how do you get to that, say, using FireBug.
When I look at either a jQuery returned object or a simple JavaScript object in the watches window of Firebug or any of the Developer Tools in most popular browsers, I see a long laundry list of properties/keys and I don't know which one to look at. In a jQuery object, most of the properties are lamdas.
So, really, my question is -- how do you know the underlying type, how do you know what's actually being returned?

The type of theElement will be [object jQuery].
If you want the HTML element itself, you have to select it:
console.log(theElement[0]) //Return <div id='theId'>
console.log(theElement.get(0)) //Return <div id='theId'>
If you want the node name, there is a property in the HTML node element call nodeName wich return the capitalised node name:
console.log(theElement[0].nodeName)// Return DIV

typeof(jQueryElementList.get(0)) will return a string of the type.
Some browsers might return this as upper or lower case, I think. IE probably uppercases (see Testing the type of a DOM element in JavaScript). Apparently you can check the nodeType attribute (jQueryElementList.get(0).nodeType) to determine whether it is an html object/tag.

Why should y.innerHTML = x.innerHTML; be avoided?

Let's say that we have a DIV x on the page and we want to duplicate ("copy-paste") the contents of that DIV into another DIV y. We could do this like so:
y.innerHTML = x.innerHTML;
or with jQuery:
$(y).html( $(x).html() );
However, it appears that this method is not a good idea, and that it should be avoided.
(1) Why should this method be avoided?
(2) How should this be done instead?
Update:
For the sake of this question let's assume that there are no elements with ID's inside the DIV x.
(Sorry I forgot to cover this case in my original question.)
Conclusion:
I have posted my own answer to this question below (as I originally intended). Now, I also planed to accept my own answer :P, but lonesomeday's answer is so amazing that I have to accept it instead.

This method of "copying" HTML elements from one place to another is the result of a misapprehension of what a browser does. Browsers don't keep an HTML document in memory somewhere and repeatedly modify the HTML based on commands from JavaScript.
When a browser first loads a page, it parses the HTML document and turns it into a DOM structure. This is a relationship of objects following a W3C standard (well, mostly...). The original HTML is from then on completely redundant. The browser doesn't care what the original HTML structure was; its understanding of the web page is the DOM structure that was created from it. If your HTML markup was incorrect/invalid, it will be corrected in some way by the web browser; the DOM structure will not contain the invalid code in any way.
Basically, HTML should be treated as a way of serialising a DOM structure to be passed over the internet or stored in a file locally.
It should not, therefore, be used for modifying an existing web page. The DOM (Document Object Model) has a system for changing the content of a page. This is based on the relationship of nodes, not on the HTML serialisation. So when you add an li to a ul, you have these two options (assuming ul is the list element):
// option 1: innerHTML
ul.innerHTML += '<li>foobar</li>';
// option 2: DOM manipulation
var li = document.createElement('li');
li.appendChild(document.createTextNode('foobar'));
ul.appendChild(li);
Now, the first option looks a lot simpler, but this is only because the browser has abstracted a lot away for you: internally, the browser has to convert the element's children to a string, then append some content, then convert the string back to a DOM structure. The second option corresponds to the browser's native understanding of what's going on.
The second major consideration is to think about the limitations of HTML. When you think about a webpage, not everything relevant to the element can be serialised to HTML. For instance, event handlers bound with x.onclick = function(); or x.addEventListener(...) won't be replicated in innerHTML, so they won't be copied across. So the new elements in y won't have the event listeners. This probably isn't what you want.
So the way around this is to work with the native DOM methods:
for (var i = 0; i < x.childNodes.length; i++) {
y.appendChild(x.childNodes[i].cloneNode(true));
}
Reading the MDN documentation will probably help to understand this way of doing things:
appendChild
cloneNode
childNodes
Now the problem with this (as with option 2 in the code example above) is that it is very verbose, far longer than the innerHTML option would be. This is when you appreciate having a JavaScript library that does this kind of thing for you. For example, in jQuery:
$('#y').html($('#x').clone(true, true).contents());
This is a lot more explicit about what you want to happen. As well as having various performance benefits and preserving event handlers, for example, it also helps you to understand what your code is doing. This is good for your soul as a JavaScript programmer and makes bizarre errors significantly less likely!

You can duplicate IDs which need to be unique.
jQuery's clone method call like, $(element).clone(true); will clone data and event listeners, but ID's will still also be cloned. So to avoid duplicate IDs, don't use IDs for items that need to be cloned.

It should be avoided because then you lose any handlers that may have been on that
DOM element.
You can try to get around that by appending clones of the DOM elements instead of completely overwriting them.

First, let's define the task that has to be accomplished here:
All child nodes of DIV x have to be "copied" (together with all its descendants = deep copy) and "pasted" into the DIV y. If any of the descendants of x has one or more event handlers bound to it, we would presumably want those handlers to continue working on the copies (once they have been placed inside y).
Now, this is not a trivial task. Luckily, the jQuery library (and all the other popular libraries as well I assume) offers a convenient method to accomplish this task: .clone(). Using this method, the solution could be written like so:
$( x ).contents().clone( true ).appendTo( y );
The above solution is the answer to question (2). Now, let's tackle question (1):
This
y.innerHTML = x.innerHTML;
is not just a bad idea - it's an awful one. Let me explain...
The above statement can be broken down into two steps.
The expression x.innerHTML is evaluated,
That return value of that expression (which is a string) is assigned to y.innerHTML.
The nodes that we want to copy (the child nodes of x) are DOM nodes. They are objects that exist in the browser's memory. When evaluating x.innerHTML, the browser serializes (stringifies) those DOM nodes into a string (HTML source code string).
Now, if we needed such a string (to store it in a database, for instance), then this serialization would be understandable. However, we do not need such a string (at least not as an end-product).
In step 2, we are assigning this string to y.innerHTML. The browser evaluates this by parsing the string which results in a set of DOM nodes which are then inserted into DIV y (as child nodes).
So, to sum up:
Child nodes of x --> stringifying --> HTML source code string --> parsing --> Nodes (copies)
So, what's the problem with this approach? Well, DOM nodes may contain properties and functionality which cannot and therefore won't be serialized. The most important such functionality are event handlers that are bound to descendants of x - the copies of those elements won't have any event handlers bound to them. The handlers got lost in the process.
An interesting analogy can be made here:
Digital signal --> D/A conversion --> Analog signal --> A/D conversion --> Digital signal
As you probably know, the resulting digital signal is not an exact copy of the original digital signal - some information got lost in the process.
I hope you understand now why y.innerHTML = x.innerHTML should be avoided.

I wouldn't do it simply because you're asking the browser to re-parse HTML markup that has already been parsed.
I'd be more inclined to use the native cloneNode(true) to duplicate the existing DOM elements.
var node, i=0;
while( node = x.childNodes[ i++ ] ) {
y.appendChild( node.cloneNode( true ) );
}

Well it really depends. There is a possibility of creating duplicate elements with the same ID, which is never a good thing.
jQuery also has methods that can do this for you.

In DOM is it OK to use .notation for getting/setting attributes?

In DOM, is it OK to refer to an element's attributes like this:
var universe = document.getElementById('universe');
universe.origin = 'big_bang';
universe.creator = null;
universe.style.deterministic = true;
? My deep respect for objects and their privacy, and my sense that things might go terribly wrong if I am not careful, makes me want to do everything more like this:
var universe = document.getElementById('universe');
if(universe.hasAttribute('origin')) then universe.origin = 'big_bang';
etc...
Is it really necessary to use those accessor methods? Of course it may be more or less necessary depending on how certain I am that the elements I am manipulating will have the attributes I expect them to, but in general do the DOM guys consider it OK to use .notation rather than getters and setters?
Thanks!

For XML documents, you must use getAttribute/setAttribute/removeAttribute etc. There is no mapping from JavaScript properties to DOM attributes.
For HTML documents, you can use getAttribute et al to access attributes, but it's best not to because IE6-7 has difficulties with it. The DOM Level 2 HTML properties are not only more reliable, but also easier to read.
It's unclear whether you're using XML or HTML documents here. Clearly origin is not an HTML attribute; ‘custom’ elements and attributes like this should not be included in HTML documents. But it's unclear what universe.style.deterministic refers to; you wouldn't get a CSS style lookup mapped without an HTML style attribute.

Yes, it's fine ;-)
If there's an attribute in the DOM, you can set it or get it, directly.
No private or read-only elements or anything. By the way, JavaScript doesn't have a 'then' keyword.

Due to cross browser issues I always use getAttribute and setAttribute:
if(!universe.getAttribute('origin'))
{
universe.setAttribute('origin', 'big_bang');
}
I don't recall the specifics but I have had problems with the property style universe.origin and dynamically created DOM elements.

No, it's not fine to do so. Most properties of DOM objects can be overwritten. You won't ruin the browser's behavior, since it doesn't use the DOM API. But you will ruin your JS scripts if they attempt to use the overwritten property in its original meaning.
My own way of doing things, when I have several attributes to attach to an object (as opposed to a single flag or link), is to create a custom object and then link it from the DOM element:
var Universe = {
origin: "big_bang",
creator: null,
style: { deterministic: true }
};
document.getElementById('universe')._universe = Universe;

How to add/update an attribute to an HTML element using JavaScript?

I'm trying to find a way that will add / update attribute using JavaScript. I know I can do it with setAttribute() function but that doesn't work in IE.

You can read here about the behaviour of attributes in many different browsers, including IE.
element.setAttribute() should do the trick, even in IE. Did you try it? If it doesn't work, then maybe
element.attributeName = 'value' might work.

What seems easy is actually tricky if you want to be completely compatible.
var e = document.createElement('div');
Let's say you have an id of 'div1' to add.
e['id'] = 'div1';
e.id = 'div1';
e.attributes['id'] = 'div1';
e.createAttribute('id','div1')
These will all work except the last in IE 5.5 (which is ancient history at this point but still is XP's default with no updates).
But there are contingencies, of course.
Will not work in IE prior to 8:e.attributes['style']
Will not error but won't actually set the class, it must be className:e['class'] .
However, if you're using attributes then this WILL work:e.attributes['class']
In summary, think of attributes as literal and object-oriented.
In literal, you just want it to spit out x='y' and not think about it. This is what attributes, setAttribute, createAttribute is for (except for IE's style exception). But because these are really objects things can get confused.
Since you are going to the trouble of properly creating a DOM element instead of jQuery innerHTML slop, I would treat it like one and stick with the e.className = 'fooClass' and e.id = 'fooID'. This is a design preference, but in this instance trying to treat is as anything other than an object works against you.
It will never backfire on you like the other methods might, just be aware of class being className and style being an object so it's style.width not style="width:50px". Also remember tagName but this is already set by createElement so you shouldn't need to worry about it.
This was longer than I wanted, but CSS manipulation in JS is tricky business.

Obligatory jQuery solution. Finds and sets the title attribute to foo. Note this selects a single element since I'm doing it by id, but you could easily set the same attribute on a collection by changing the selector.
$('#element').attr( 'title', 'foo' );

What do you want to do with the attribute? Is it an html attribute or something of your own?
Most of the time you can simply address it as a property: want to set a title on an element? element.title = "foo" will do it.
For your own custom JS attributes the DOM is naturally extensible (aka expando=true), the simple upshot of which is that you can do element.myCustomFlag = foo and subsequently read it without issue.