GFCI outlet in cabinet above under-cabinet range hood
Is it code-compliant to install a GFCI receptacle inside a kitchen cabinet above the range hood?
Yes. It's commonplace for over-the-range microwave range hoods, and by extension would be acceptable for more conventional range hoods as well.

Does the outlet require any sort of special enclosure?
No. A standard electrical box and trim plate will do.

Can the appliance power cord just sit there, bare, at the back of the cabinet, plugged into the GFCI outlet?
Yes. The kit will often include a D-loop and screw for mounting the extra cord against the wall.