as far as Obama asking for the drone back, that could be done for a variety of reasons, but one that I can think of is to see how and if whatever available HUMINT resources available in Tehran jive up with whatever ELINT information we gather. basically a quality control measure of intell gathering.
Obviously, if it is a fake, they won't be loading it up on a trailer, and driving it to the border so it can be handed over.
and if we never really did lose one in the first place, we could get to see what assets the Iranians would use to go search for a drone that isn't there.
Say like a month from now, we could release pictures of a new drone entirely, call it the RQ-180 Stingray or whatever, and then in March say that we lost one just to see what or how Iran interacts with Russia, China, and North Korea.