The number of enemies like those are infinite, though it can take some time till new ones show up.
When you kill a "nemesis", you are awarded with a rune. Those runes power up one of your weapons.
Naturally, a dead nemesis leaves an open spot in the commanding ranks. These ranks get filled over time. Certain other events also can fill up the ranks (like if you get killed by an, yet unknown uruk, he will be promoted).
There is no drawback in killing those. By killing them, you gain power and reduce the strength of Sauron's army. You also weaken the highest ranking uruks, the warchiefs.
Before you kill them, you should interrogate them (if you get the chance). They will spill useful information (like weaknesses of other named uruks). Later in the game, you will be able to even recruit them for your own purposes.
Warchiefs are pretty powerful and on the top of command. They control 2~4 named guys under them (captains). They also act as bodyguards. You won't encounter them on the map, they must be drawn out by having a dominated nemesis bring him out or completing a mission that will attract the warchief's attention.
EDIT:
After playing the game for a bit, I read that the more powerful a nemesis gets, the better the rewards are for slaying them. So, you can somewhat "train" them (by letting them kill you, for example).
Source:
Gamesradar article
IGN - Nemeses System
When you die, there is a chance a "bounty" will be placed on your killer:
This is available for anyone in the game. Even randoms. So by default, if you ever see any of these, you will be avenging the deaths of friends by killing the target.
Best Answer
I've been playing the game with two friends and from what we can tell it is only in your game that the general will die. More often than not, the vengeance mission will be against a captain that your friend has already got revenge on too.
But to reiterate, no general ever dropped dead on me when my friend got vengeance on them for me.