I dare say I betray my enduring mechanical ineptitude here, but alas:
When I push the bike, I hear a scraping sound from what seems to be the brake discs (possibly the wheels themselves?). Is this normal? The wheels turn a few revolutions if you spin them so it's obviously not putting that much force onto the wheel. Does something need cleaning / lubing / replacing?
Perfectly normal. Brake pads are always in minor contact with the rotors, hence the scraping sound.
There would only be a problem if the brakes were dragging excessively; because you said your wheels will spin a few times on their own, this does not seem to be a problem in your case.
:)
+1