If Taki hits with a multiple will it go face down in my staging area at the end of the turn?

Only if some effect would make it do so. Like, for example, her enhance. It is however an enhance, not a continuous ability, so just playing as Taki is not sufficient on it's own.

So even if you use the Taki E on a multiple you would still have to discard it at the end of the turn.

Um, no. That's the opposite of what I said. If you play the enhance, and the attack deals damage it will come down at the end of turn, multiple or not.

Aslum's just being excessively wordy. Yes, if you use her Enhance on a multiple copy it'll come down as a foundation.