1. Absolutely runs in a browser and we're doing all we can to make it a first-class citizen. The idea is that you can bring ad hoc members to the team without them having to do any setup.
2. We're starting to build out native apps with deeper integrations (ie, the Mac notification center). We're starting with the commercial desktop and mobile platforms, but as a big fan of Linux I'm eager to have a native client there, too.
3. Right now it's a hosted service, but others, too, have brought up the sensitivity of team data. Reminds me of how Memoto received feedback that eventually brought them to incorporate an option for local storage rather than pushing images to their servers automatically.
4. At this early stage it's a custom protocol, but that was the fastest way to stand up our interface layer. I do think it would greatly help adoption to be XMPP compatible and we're digging in to the potential of that. Video / audio (which is used to complement the chat) is based on WebRTC.