1 - You don't need a dedicated server if you plan on using the built in multiplayer plugin since it's based on peer-to-peer technology and uses Scirra's signalling server. If you want to, you can choose a different approach. Some of the developers here made custom servers for their games with Node.JS. Keep in mind that multiplayer is one of the hardest features in a game so I'd recommend some testing before you jump into any big projects.
2 - Can't say anything about that.
3 - Maybe with some 3rd party plugin. I'm not sure this can be done with the vanilla C2.