You could make the game you want with C2. It all comes down to time and investement. In some cases you are going to have to do a lot of ground work to fortify the parts that C2 doesn't cover. Custom plugins etc. But likely not so much.
I like you am designing a overall game, by developing it in small parts. Good luck :)
As for WebSockets. You can make Realtime game play with Websockets just fine. However, you need to keep in mind idea usage and expecations. TCP is not ideal for high paced competitive gaming. Mostly shooters(Call of Duty, Halo) and Fighters(Street Fighter). This only really applies to competative gaming. In non compatative gaming you can usually get a way with it. And games that aren't high twitch you will be ok too. I month and little bit ago I wrote a websocket top down shooter in C2. Played fine as long as the ISP and connection was good. Played people from West Coast Canada vs European players. no problem. yet on occasion someone in the same city would be laggy as hell :|..
in an ideal world. TCP is no different than UDP. TCP however has data safetry protocols. And when data is lost, or fragmented TCP attempts to fix this. The problem is that TCP forces an in order handling of data. So even if TCP get's data, it will hold on until it fixes a lost message.
so UDP is better, for twitch games due to this. As often game messages can ignore lost messages due to the fact that some of the same data is in the next message. and more up to date.
but you can still do online gaming fine.
as for what tools you have at your disposal
1. Scirra WebSocket plugin.
This will require you to build the entire infrastructure for your games. A lot of work, but you get what you design.
2. Rexrainbow's Bottleneck. This is fairly fleshed out, and if you have programming talent. You can do the server logic in JS for NodeJS. However, bottleneck already covers rooms, connections, disconnects, join/leave... so on so forth.
3. PubNub, a recent discovery is that PubNub has an old out of date C2 plugin. However, if people use it and prove a community for the PubNub plugin. They might end up supporting it more effectivly.
4. VelotJay networking plugins, he has a few. A socket.io plugin and a photon behaviour. I'm not sure how much they are fleshed out for practical gaming.but i'm sure you can always tinker it up :)
5. My unfinished photon client plugin. It mostly needs to have the loadbalancing features added. ie room lists. however, I did have a shooter game using it.
6. And then for those with patience. With awarmenhoven and I are working on a network game api. While it's a JS api, it's built with C2 developers in mind. So the feature set is to be easy to use. but again. that's for those who are patient.