Can I move plot-specific buildings ?
Is it possible to destroy, then rebuild plot-specific single buildings, such as the alien containment or officer school?
Will I lose anything?
Yes, you can. You will only lose money and building time.
In particular, destroying and rebuilding alien containment is a way to kill all aliens in it, so you'd have their corpses for the Grey Market or Council Requests. This is the only way to kill of duplicate aliens, as you can interrogate particular kind of alien only once. Duplicate aliens just sit in the Alien Containment forever, providing no added value whatsoever.