does the NEC allow conduit attached to my house to an outbuilding or does it have to be buried?
Yes, it's permitted in Article 225. But...

If exposed to temperature extremes, you must provide "expansion fittings" per 300.7 and as a general rule, running it along the side of a building wall usually qualifies as exposure to temperature extremes.

Article 352 ALL pertains to using PVC conduit, but there are a couple of things to note:
A) 352.12 says that you can't use it where subject to physical damage, but...
B) 352.10 had just said that you CAN use it. The subtle and often missed detail is that 352.10 has a note explaining that SCHEDULE 80 PVC is allowed for use where physical damage is probable. If you are unsure of how your situation applies, just use Schedule 80.