Try team death-match. it should fulfill your needs.
No, because team deathmatch uses the ‘reinforcements’ system (which, by the way, doesn’t seem to be listed anywhere in the default UI so it’s impossible to tell what your team is at) which means that when a team gets rolled, half of the people get to sit in spectator and watch the last survivors run around for many minutes until they’re caught.
Also, there are many bugs with the reinforcement system.
For example, tonight I played a TDM game where the opponent ran out of reinforcements. I personally killed the last several members of their team still alive (with a nice 3 k/d ratio if I do say so myself) when the dead members of their team who are forced into spectator leave the game en-masse (and understandably so, as it’s boring to sit in spectator and watch the game trickle out). Then, even though the entire opposing team was dead, I was autobalanced to the opposing side and forced to spawn and be killed, then greeted with a losing message - even though moments earlier I had killed the last reinforcements of the enemy team, and they were all dead. The autobalance system ignored the fact that the enemy team was dead and the game should have ended, and instead forced a bunch of winning team members to be balanced over and killed.
Problems like that wouldn’t exist in a game match that was a race to X many kills instead of a silly reinforcement system. It would also prevent steamroll matches where one team picks up steam and obliterates another due to sheer numbers, what with the limited spawns. The current TDM model both allows for very one-sided games as losing teams simply become outnumbered, as well as a lot of boring waiting in spectator if your team is losing. I don’t see how that’s fun at all.