Great stuff @Devanil, thanks!
Responses to some concerns:
The sidebar's really rough right now, I agree. In my head I was thinking the following about it:
We add BabbleSidebar and BabbleDropdown site settings which allow you to pick whether the chat appears on the navbar, the sidebar, or both. Potentially could add support for putting it in other places as well, like above the topic lists as you mention? (Personally I think having it there would be too prominent for me, but I don't think it's a massive effort to support it.)
On mobile (ie, at a certain minimum screen width), we do some css-fu to hide the sidebar and replace it with an icon that you can press to make the chat full screen.
Good feedback on the user name and enter to submit, those are easy wins I'll put on the TODO list, and you're right that chat shouldn't affect user post numbers.