Hunters curse only affects them if the hunter is killed by it. Doesn't matter how the vamp kills the hunter though.
Curse continues until another potential hunter makes his/her first kill. There must always be 5 I think.
I think Damon was bluffing or just being a bad *** when he threatened to kill his hunter. Didn't he make comment that 'after he becomes human' he'd kill the hunter?
ETA - I don't think Damon killed a hunter. Elena killed the first one we saw. Klaus has killed one and had to suffer the torment of the curse for 50 years.