It does not need to match your entire team's race. The only fundamental requirement to even gain access to a stage depends entirely on your team only | . The process of selecting a companion comes after that so it's only logical (to me) to go off based on the same principle. There'd be very little reasoning to restrict people by doing otherwise.
Yes, of course. The original rule states "as a leader or companion". In the case you were to use them in either of the aforementioned positions (even if it's against the rule), you could provide evidence to prove you never used the monster's skills at any moment and under these circumstances, the only way to prove this is through a video.