i'd name him Hiro, not for any aspect of the fish, but because you were a hero for rescuing him! kudos!
as for the idea that his fin damage is from tail biting, i'd be inclined to agree. living in that little cup he had nothing to keep him occupied and his body was most likely curved most of the time due to space constraints, so he probably saw his tail all the time and bit it. either that or it is fin rot, but the edges are fairly clean, so i'm hedging on the tail biting.
and to answer the 'melafix is not good for bettas' question: melafix (and bettafix too!) contains melacula (spelling?) oil and that is not supposed to be good for labyrinth fishes, which a betta is.