Logically, it wouldn't make sense for Luke Skywalker to be in play while Red Five is also in play. Even a Jedi can't mow down stormtroopers with his lightsaber while also piloting an X-wing in outer space. But I don't see where the rules would specifically prevent this. Luke Skywalker is one unique card, and Red Five is a different unique card. So technically, both of them could be in play at the same time.

Same question for Darth Vader and Vader's Advanced Tie. Are both unique cards allowed to be in play at the same time, even though Darth Vader can't be in both places at once?

Both of those sets of cards can be in play at the same time. Later sets (ie the Rogue Squadron cycle) expand upon the "pilot" mechanic introduced with the Black Squadron Pilot and allow you to add specific characters as pilots to vehicles. This includes pilot versions of both Luke and Darth Vader.