ok one of the things that really really infuriates me about PvP is the fact that you just go in guns blazing and care about absalutely nothing cause when you die you know you just get respawned in 30 seconds later with everything you had on there and no penalty whatso ever, there is no thought of keeping yourself or your team mates alive .
so i propose a death match mode where once your ship goes down, your down and out but you get to watch the rest of the match in a spectate mode or something, this would be an awesome tool for fleet vs fleet matches and encourage ALOT more teamwork that does seem almost non existent in fleets these days.
The word you're looking for this game type is Last Team Standing. The current Arena model is what would generally be called Team Deathmatch, and Deathmatch refers to the same thing in an everyone-for-themselves fight, as opposed to teams.
Fights in STO are already team-centric tug-of-war matches that snowball rapidly as soon as something (generally a precisely orchestrated timing attack) breaks the the balance. There is virtually no way to come back from a 3v5 or even a 3v4 in this game, simply because of the nature of the push-and-shove of heals, resists, damage, and debuffs. I think that a last-team-standing model would not work very well in STO.
I also think it wouldn't solve the problem you're describing. A good team already supports its teammates with cross-healing, defensive debuffs and controls, etc, and will crush any team that doesn't. Pug teams often do this too, it's not just a premade thing. All that making it last-team-standing would do is make fights much shorter, and extremely frustrating for the unlucky person who pops first.
Following the shooter archetype, though, I think some of the following would be crazy good fun and would work for both ground and space.
Capture the flag
King of the Hill
Zombie mode (players start out on the same team against NPCs, and end up on the NPC team as they die. Champions Online has this and it's great. Zombie -> Borg, die-> assimilated.)