1. The IDE is designed to display only events.
Nevertheless you can use the SDK to code your own plugin/behavior.
Make sure first that you're not reinventing the wheel and that either the functionality you're looking for isn't already in C2 built-in extensions, or if another custom plugin/behavior hasn't already been released on the matter.
Having a look at this tutorial might help you getting the global C2 architecture.
2. The how do I FAQ has several elements listed at the section "AJAX" which should answer this question.
Be sure to also check AJAX's manual entry to see about the last features implemented.
For the node.js server there are a few multiplayer tests that have been thrown around, you can see some listed in the section "Networking".
3. The tutorial "how to make a platform game" should display you the way to import animations in C2's sprites.
Some behaviors like "Sine" can help you giving some animation illusion too, but as for vectorial animation, there's nothing of the like implemented in C2. But I guess it's all about moving objects around, with bullet behaviors for example (you can check the section "Object moving" in the how do I FAQ).