Once the aspect exists, it exists (unless it is not sticky, which is a matter of duration). Unless someone else acts to remove the aspect.
So, yes, you do not need to give your tag away.
Yes, consequences are aspects, but, "saw my friend killed by a demon" is not that great of a consequence. A consequence should be something that affects you "spooked by a demon", "obsessed with the demon that killed your friend" "Nightmares about fred being eaten by a demon". For physical consequences "punched in the face" is not a good consequence, compared to "sore jaw" or even "man, those are some massive fist marks on ya"
I also remember that fate points for being invoked against show up at the end of combat, so you don't have the same fate point(s) going back and forth each action.