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Javascript framework for draggable objects on web browsers

What would be the best Javascript framework to implement objects that can be dragged around the window and hovered over to make other objects appear?
I was thinking about AngularJS but is that the best choice? jQuery doesn't seem to be versatile enough.
Also, which one is the most cross-browser compatible?
Thanks
Edit: jQuery UI could be an option but it seems that what I am trying to do would be animation-heavy and a framework using hardware acceleration would be more suitable than one using browser acceleration?

You're only going to get native acceleration using HTML5 in combination with CSS3. To that end, there are means of implementing something similar to dragability, which, in combination with this answer, may achieve what you need via the events. JS still required, but much, much less.
The downside is that you lose browser compatibility. To remedy this, check out Modernizr. You can add in jQuery UI if a legacy browser accesses your site, but otherwise stick with HTML5. As you can guess, doing animations without Flash and with native acceleration AND browser compatibility can get hairy quickly.

What browsers support interactive SVG documents?

I'm working on a project that requires an interactive map. I want to use an SVG document with some custom javascript code embedded in it much like this developer has done:
http://treeblurb.com/xmap/svg/melbourne_central.svg
Achieving what I want to do is trivial but I can't tell which browsers would support this level of interaction. Essentially I need to:
Embed custom script tags into the document.
Adjust attributes of nodes (i.e. fill)
Add mouse events to specific elements.
I guess my question is - if a browser supports native SVG, would it allow me to do all of these things?

Native SVG is supported by most versions of Chrome, Firefox, Opera and Safari. IE8 does not support it, but IE9 and above does. Android supports it from Honeycomb onward.
You may want to take a look at Raphael. It not only makes SVG-like functionality available in IE8 and below (by falling back to VML), it also provides methods to manipulate all objects, which you can even combine with JQuery.
There's also good information comparing Raphael to JQuery SVG here.

I.E. 8 and earlier does not support SVG. Android 2.3 and earlier also does not support SVG. Check out more browser compatibility issues here: http://www.caniuse.com/

Javascript animation: IE vs other browsers

Before I launch into the specifics of the issues I am facing, I just need to ask: Is it a mistake to use IE8 as the reference for building a website, particularly one that uses JavaScript animation? I ask because I have written a fairly simple animation page, mostly from scratch, even with my weak grasp of HTML and JavaScript languages, using IE8 to monitor the progress. I have tweaked the code so that it works just fine in IE8 (compatibility mode turned off), but when I tried it in Safari and Chrome, it does some weird hiccuping in the animation.
I find a lot of questions with the opposite problem: that it works well in everything but IE. So I am wondering, should I be using a different browser for my reference? Or is there a better approach to make it compatible with all browsers? It's so frustrating (as I am sure most of you will agree) to have to deal with the different interpretations of the different browsers.
Thanks for any help!
p.s. I have not coded yet for Mozilla.

Using ie8 as a reference from a performance point of view is not a bad idea, since it has weak js and rendering performance. Using ie7 is even better.
Cross browser compatibility wise, it doesn't matter what you use - unless you use a crossbrowser library like jQuery as your base for the animations you will have to write specific code for the various js and render engines.
Even if you use a library like jQuery, you will still run into rendering issues since the various html/render engines are different across browsers. chrome/safari uses webkit, firefox uses gecko etc.
The only way to do it right is to start your project by defining what browsers you wish to support and then test what you do in all of them while you are developing your code.

If you're doing animations, I encourage you to take advantage of css transitions instead of controlling the elements via javascript - you'll have an opportunity to take advantage of graphics hardware as well as more efficient drawing of pixels in general.
If you need to still perform animations in browsers that don't support those css transitions ( some in ie9, none in ie6-8) then you can use a tool like modernizr to detect what's available and control those elements in the event those features aren't available.
Typically, the animations are extra - so I've had good success in ditching the animations for IE users - if you're making slow, javascript driven animations just to work for IE users, you're punishing users that are using better browsers.
Just my $0.02 of course

Add style elements in CSS

I want to make my own custom CSS elements that will be handled by a js file of mine. Is this possible?
Here would be an example:
div {
rounded-corners:15px 15px
}
And then use Javascript to apply the styles for each browser for rounding corners.
Is there a way to do something like this?
EDIT
The point isn't to add a common support for browsers. I want to implement my own CSS things.

The answer is "no". While it is possible to easily get style values from stylesheets in Internet Explorer by accessing the currentStyle property, like so:
alert(myDiv.currentStyle['rounded-corners']);
//-> "15px 15px"
...other browsers don't support the currentStyle property, opting instead to support the W3C standard window.getComputedStyle(). getComputedStyle() doesn't include "expando" style properties in the outcome, which means your only option would be to iterate over the rules in each stylesheet instead, which could be an expensive procedure depending on how many stylesheets and rules you have. Obviously, you also lose out on the browser's cascading/computing.

IE9 and the latest versions of Firefox and Chrome all handle border-radius anyway.
And you can use CSS3 Pie to add support to older version of IE. No need to write your own script.
EDIT: I suppose you could write your own version of CSS3 Pie, but why would you want to?

You may want to look into SASS/SCSS or LESS, which both offer "mixins". These act like functions in CSS, and should let you do what you want. LESS at least can be made to work client-side through Javascript; I suspect SCSS can too.

CSS already has a property for rounded corners, called border-radius.
This is supported by virtually all browsers in current use. The only exception of any significance is IE8 and earlier.
The good news in this case is that IE8 can indeed be programmed to work the way you're asking. Other browsers cannot, but IE can, and this is lucky, as it's the one browser that needs it more often than most.
So the direct answer to your question is "sometimes". Most browsers drop the styles they don't support, but IE keeps them. It obviously don't do anything with them, but you can access them via the DOM, which means that you can do what you're asking.... but only really in IE.
In the case of rounded corners, there is already an excellent Javascript-based hack for it called CSS3Pie. This is open source, so you can examine their source code to learn how it's done.
The CSS3Pie code is quite complex though, so if you want a simpler example to work from, take a look at this older script which does the same thing a lot more simplistically. Easier to read, but not as functional. For real life use, use CSS3Pie, but for learning, this one is a better starting point.
For working with IE, I would suggest following the examples in these scripts to achieve what you're asking. For other browsers, you will have a much harder time.

How to give CSS3 support to IE?

I use IE7.js but it doesn't have CSS3 support. I use jQuery always in my projects.
What is the best lightweight way to give all CSS3 selectors and properties support to IE 6,7,8?
I'm not asking for HTML5 support only asking to give CSS3 support in as much as light on performance way.

Try keith clarks IE CSS3 pseudo selector emulator: http://www.keithclark.co.uk/labs/ie-css3/
I haven't personally used it but it manually parses your css file and adapts it for ie-browsers.
A downside is you're relying on js for css functionality..

the IE7.js project linked in the question has recently added IE9.js. This brings more CSS3 selector support to legacy IE.

What your looking for is this new library: CSS3 PIE
http://css3pie.com/
It gives IE6-8 several CSS3 attribute support.

There is a thread on this running on webmasters.stackexchange.com with a lot of projects listed:
https://webmasters.stackexchange.com/questions/2350/how-to-enable-css3-features-in-internet-explorer-6-7-and-8/2397#2397

You can't really change how a browser implements in rendering process or featured contained in that renderer. IE9 is supposed to support many (all?) CSS3 features, but you will not get to use it for some time.
Like you said, you will need a third-party tool for this.
You might be able to use Google Chrome Frame, but I am not sure if that supports CSS3 yet.

You can use Google Chrome Frame http://code.google.com/chrome/chromeframe/
This will make your IE render and work as Chrome, which supports all CSS3. All you have to do is to add this <meta http-equiv="X-UA-Compatible" content="chrome=1">
after your <head>. You can try it out online at http://www.typefolly.com/
You have several CSS3 options there like text-shadow or transform.

Following is a good workaround to use ccs3 properties in i-e 6 to 9
http://css3pie.com/documentation/getting-started/