Everything sounds right here. But, the orcs and Uruk-hai are not immortal. The original elves, and by that I mean the very elves who were captured by Morgoth, decided to give up their immortal lives because they were becoming so evil and impure, so that if they did live forever, they knew that they would be used to wreak havoc on Middle-Earth. Their "children" were not given an immortal life as they were no longer elves and their "parents" had no immortal life either and so on and so on.
This is all very complicated but I'm sure it says it all in "The treason of Isengard." But I'm not sure, I just thought I'd throw that in there.
