I don't really understand what your issue/question is but:
Players can join servers mid-game normally. Unless you mean to force them to play right away instead of being spectators for a round? Then that's very likely a GSC script that you need to find or create
I didn't experience maps ending when you're solo. Are you sure it doesn't only happen on specific maps?
If you could give a clear/precise case of what you're saying that would help because I have no idea what you're talking about, never saw that myself