No. Identity Thief doesn't enter the battlefield again when it becomes a copy of something. To enter the battlefield means to be put onto the battlefield from some other zone, and Identity Thief has been on the battlefield the whole time.
And yet another 5 life when it returns to the battlefield at my end step?
Yes, it sees that a permanent you controlled left the battlefield this turn so it triggers again on entry. (You don't even have to have sacrificed the Map for this one - it would trigger just because Aeronauts itself left the battlefield earlier.)