Develop Reference

AJAX - How can I build a notification system, that is constantly getting updated, without slowing down my website too much?

I am a beginner to web development, and I am trying to do a notification system with AJAX and jQuery.
In my web application, I have a comment system where you can mention another user. After a comment mentioning a certain user has been written, a new entry on my notifications table will be added, containing the comment, the id of the user who commented and the id of the user(s) who will receive the comment. After the notification is stored in the database, I want the person that was mentioned to receive the notification.
To that effect, I decided to use AJAX. Using the setTimeout() method, I am sending an AJAX request to the database every 2 seconds, and with that, I can display the notifications visually to the user that is meant to receive them.
My only concern is that this will slow down the site once I connect it with a server.
So, I was looking for a way that would allow me to implement a notifications system without slowing the site too much, since the one that I am using currently doesn't seem very efficient.
I would appreciate any help.

Implementing ajax to make a functioning live chat

I'm trying to build a simple live chat for my website, I already have everything set up, meaning databases, login and log out system, forms, displaying the database values in chat form.
The only thing I'm missing is implementing the live refresh function so users don't have to manually refresh the site to see new messages.
I'm a little lost, could you give me some guidance?
I'm guessing I have to add a specific javascript call whenever a message is sent, or the "send" button is pressed to be more specific, then that would call back a function to refresh the chat for every participant. Not sure where to start or which code to use.

Essentially, you have 2 ways to implement this.
Using a polling technique with ajax request which will grant you a greater compatibility, but you will have to much more work to implement it. I don't really consider it for most cases nowadays, because it could be very inefficient.
Using a socket mechanism, you can use web sockets for a out of the box solution, most browsers have support for it nowadays For third party libraries you can use socket.io which can fall back to web socket yet it grants a little bit for features out of the box, like channels, which you would benefit from. Lastly, for a third party service, you can use firebase/firestore which they have a realtime database, so whenever a change happens, you get notified.
I would recommend using either the web socket approach (native) or using a wrapper/library like socket.io. If you go with Socket.io, there are a lot of tutorials out there that build chats with that library, so you can get a working sample really fast.

Updating ruby on rails app with jQuery every time there is a change

I Want to make an app with ruby on rails and Jquery that will allow multiple users to have the same page open at the same time, and if any of them makes a change to the page, adds a post, or deletes a post it will show all other users that change without having to reload the page.
Here in stack-overflow, whenever another user comes and gives me a point or removes a point on the post, it will show me without having to refresh the page.
Same with the comments, if someone posts, I will see it without having to refresh.
Can anyone tell me here to get started with this?
I would rather not have to have the page reload every 30 seconds.
Any help would be appreciated.
Thank you in advance.

There are a few ways to go about this:
Websockets
Server Sent Events via ActionController::Live (Rails 4+)
Long Polling (outdated method at this point)
Between websockets and SSE I would go with the former. Higher browser compatibility and the more mature technology of the two. If you're willing to pay for convenience, check our Pusher (solid free tier). Otherwise you might want to check out something like Faye (good intro at http://railscasts.com/episodes/260-messaging-with-faye).

Edit HTML of active Facebook page (chat specifically)?

Okay, so I'm sure with all the HTML/JavaScript masters on here someone will figure this out.
Ive got this app I'm working on. The objective is this...
1) The user opens my app (objective-c/cocoa) and selects their preferred language.
2) My app opens Facebook.com in whatever the users preferred browser is, except changes the document, in the way that all chat messages received are translated to the preferred language...
How might I go by this? Pretty much replacing the received chat message strings with new ones automatically?

If you need access to Facebook chat specifically, they expose an XMPP API that you can work with: http://developers.facebook.com/docs/chat/ . I haven't used it personally, so I'm not sure if it fully satisfies your requirements, but it may be worth a look.

Implementing web chat, how do I get typing status?

Can someone illustrate how I can get typing status of the other party with JavaScript?
UPDATE
Can someone recommend a one-to-one open source chatting application, preferably written in PHP? I only found open source chatting rooms which are for chatting among all onliners, but I just need a one-to-one chatting.

Here are a list of PHP-based open-source instant messaging software.
Some of those might be relevant for you.

For example, if you had an text area #chat then you could use this code to attach the event:
document.getElementById('chat').addEventListener('keydown', FUNCTION HERE, false);

See http streaming and some ready solutions here: http://ajaxpatterns.org/HTTP_Streaming
this is how google talk does it. And there are ready php or c++ solutions
It was quie a discovery for me!

This is an update to reflect the significant change in the OP's question:
Google Chat and Facebook both use XMPP(jabber) servers, as do most companies I know of that have internal instant messaging.
The nice part about XMPP is that you get all of the "is typing" and other presence-based information without having to roll-your-own in javascript (bear in mind, you will still need to use javascript to pass XMPP requests back to the server, but XMPP has most of the features you'd need already built in).
Check out OpenFire. It's a great XMPP server, totally open source, and they have a web-based version of their Spark client that is pretty nice.
Or you could get a PHP library for XMPP (there are a few). But you'd still need to have the XMPP server running in the background for PHP to work with.
Here's a list of XMPP libraries for PHP from XMPP.org:
Eiffel
JAXL
Lightr
Missus
xmpphp
Or, if you want to keep things mostly browser-side, they also have a list of libraries for javascript:
dojox.xmpp
js.io
JSJaC
strophe.js
xmpp4gwt
xmpp4js

I made a small chat application a while ago, and the only way to do it is to frequently check for new entries in the chat database and fetch anything newer than the last displayed message. At the same time as all that, you can check to see if the user's input is empty. If it is, do nothing. If it isn't, enter a status code into the database beside that user's name. If anyone has that status in the database when you're fetching information about new messages and who is online, you should display the 'user is typing' message. I hope that makes sense...let me know if it isn't.

For User1: If you save the chat message on each key-press to the database, with a status: sent=false and update the last updated date.
For User2: you could pole periodically for the presence of a message where sent=false and use the last updated to update user is typing message. if the lastupdated date is more than a say ten seconds you could remove the message as that person may have stopped typing. This will allow User2 to see User1 typing, stopping and continueing again.
Ideally polling for this information will be part of an existing call to the database to reduce additional overhead.