For a 5+ player game, the first two players to roll fists/claws can enter Tokyo. If there is already a player in Tokyo City, but Tokyo Bay is open, does rolling fists/claws damage the monster in Tokyo City as well as allowing entry to Tokyo Bay or do they only get to enter without dealing damage?
You deal damage to the Monster in Tokyo City, then take control of Tokyo Bay (or Tokyo City if the damaged Monster abandons it).
From the rules, page two far right column:
If Tokyo Bay is unoccupied but Tokyo City is, the Monster who attacks Tokyo deals damage to the moster controlling it, then takes control of Tokyo Bay - or Tokyo City if the damaged monster abandons Tokyo City.
Yes, attack damage is resolved before checking if you move into Tokyo (regardless of whether or not Tokyo Bay is available), so the monster who went into Tokyo first will take that damage. For example, if the monster already in Tokyo wished to yield, they could do so, and the second monster would go into Tokyo City instead of Tokyo Bay.