Can I degrade back to 3.7.1 from 3.8 by just coping old wp files?

Short answer – who knows. Will probably work, maybe something will explode. Behavior is undefined for backwards direction.

Long answer – from quick look at code 3.8 upgrade doesn’t seem to do anything specific to database, so likely it will be pretty painless. For any serious site – make a staging copy and test it thoroughly.