Show user activity and readiness status in rooms

Rooms should show whether users are active, idle, competing, spectating, waiting, or currently solving.

Feedback source:
- User position for what they're doing.
- Show if people are active.
- In users online, you should be able to challenge someone, or invite them to a room.

Acceptance ideas:
- Display per-user room state in the room UI.
- Distinguish at least active/idle/spectating/competing/waiting/skipping if available.
- Use status information to support better leader transfer and host controls.