A: The host needs to pick the random player, and relay the player chosen to all peers.
B: The peer triggering/picking the random player needs to relay to the host which player was chosen, and the host needs to relay that to the other peers.
Your second question is the same. Make sure the host determines what is correct, then sync with peers.
Make sure you actually walk through and DO the multiplayer tutorials, they will make your life a lot easier.