Welcome to Halo's shitty instructions.
Swat is no shields, usually with the DMR rifle, and I think no radar.
A single head shot (or 1 billion body shots) is all it takes to kill.
Skulls? I think there are skulls you can find in single player that work like cheat codes. I think they're in Firefight mode too.
If you mean MP Headhunter. Shoot someone, take their skull, you can carry up to 10 skulls at once. To get points you must take them to the impossible to find, grey colored goals (which will move as soon as you're 5 feet away. RAGE!) before you are killed and drop them.