[Kalimac]

Very good question. I'd never gotten the impression that Orcs were loyal to anything other than their own appetites.

For "avenging a fallen captain" it probably depends on the situation. While it's difficult to see an Orc pursuing an enemy solely out of loyalty to a dead captain ("You killed my Shagrat. Prepare to die") it's easier to see them using the dead captain as a pretext of some sort, for example it's easier to see an Orcish raiding party nicking a few horses off of the Rohirrim, all the while snarling about avenging their fallen comrades who were killed by Eomer and Co. The only thing is that it seems like a lot of trouble to go through for no reason; it's not as if Rohan and the Orcs have some sort of diplomatic relationship where this sort of pretext can be used as political leverage.

There is one situation where it's easy to see them avenging a fallen captain, and that would be in cases of infighting. If one of the Rohirrim kills the captain of a band of Orcs, the Orcs would probably, as you said, just push him said, engage in a little scrapping to elect their new captain, and go on as before. But in a case like that of Gorbag and Shagrat, where you have two rival Orc companies fighting with each other, it seems believable that one of the deceased Gorbag's company might try to do in Shagrat. Not to avenge Gorbag, but to protect himself, since any of Gorbag's subordinates would probably not fare too well under Shagrat's command. It would be an act of simple self-protection, but the immediate reason for it would be the captain's death, so in that case you could, at a stretch, refer to it as avenging a dead captain even though that would be more of a side effect than anything.