Do my dying soldiers live if I end the mission before stabilizing them?
Say that I have a soldier who is dying, and I get the message saying that I have three turns to stabilize them or they die.
If I finish the mission before then, do they live? If I don't have any medpacs handy, is it worth trying to rush to finish the mission in the hopes that they'll fight another day?
Yes, they'll survive.
If you're sure you're on the final group of aliens, finishing the mission early is a valid tactic for saving them. Trying to rush the mission is also a great way to get more soldiers killed of course, so bear that in mind.