I'm playing as Noise. I put Medium in play. R&D trashes top card. Does it go face down or face up? Does Corp even get to look at the card first?

Trashed cards go to archives faceup if they were either rezzed or revealed at the time of trashing. The corp can look at facedown cards in Archives at any time.

In the specific case you asked about. Since the top card of R&D is facedown, it goes into Archives facedown. The corp can always look at cards in archives whether they are facedown or faceup

I covered this. The top card of R&D is, necessarily, neither revealed nor installed in the circumstance of the question.