It seems to me that while Google Hangouts might have been the obvious choice last year, WebRTC might actually work better now. See talky.io for a live demo. This page details how you can roll your own video chat with <20 lines of totally-front-end code. Since it all runs front-end, the cost to the server is no greater than a normal text-based chat client, as far as I can tell.

For a simple chat solution, there are lots of other libraries (I'm familiar with ShareJS for node, which makes this super easy). Honestly I think this would probably be simpler than trying to integrate with Google Hangouts, and it also makes your two "I don't see an obvious way" issues less issues (still need solving, but there are straightforward solutions) and makes the multiple rooms thing quite easy.