Apparently, NEC 210.52(D) allows a GFCI receptacle output to be
"installed on the side [...] of the basin cabinet" 
which is exactly what I would like to do in my bathroom.
Specifically, I am wanting to relocate a receptacle (on a GFCI circuit) that is currently in the wall above & behind the countertop. The way I imagine this working is: I would remove the existing gang box, and thread the existing NM cable back through all the studs it currently passes through to get there -- at most 2 feet -- then have it turn down about 1 foot before emerging out of the wall (through a hole in the drywall) and into a hole in the back of the basin cabinet.
This raises 5 questions/concerns:
I don't think I have ever seen a circumstance where NM cable just poked out of a sheetrock, to head into the back of a cabinet. I'm not certain I would be breaking any rules by doing this.
By my imagination-math, the existing NM cable would extend out of the wall by only 1 foot (drawn backwards 2 feet, but then down 1 foot, so can only protrude 1 foot), which is not enough length to get to where I want to place the receptacle. My basin cabinet is 2 feet deep, so I need at least that much slack to make the connection. I think that perhaps the best option is to place a new gang box in the wall at the 1-foot-lower location, inside which I can use wire nuts to attach a second length of romex that would come out of the wall, perhaps through a blank cover-plate that I've drilled a hole through? Do they make gang cover-plates for this type of application?
If I did #2, I'd be hiding the gang box behind the installed basin cabinet. Is that allowed?UPDATE: A commenter has indicated that making it inaccessible is not allowed. But I think if I cut a window in the back of the cabinet, then I should be OK on this detail.
I don't know what kind of strain relief I should employ at the various interfaces. Maybe at the least I should use rubber grommets both in the hole I drill in the wall's blank cover-plate, and also a grommet in the hole I drill in the back of the basin cabinet?
Do I need to consider that a future person might decide to yank this cabinet out in some future remodel, and not be mindful that there is an electrical cable heading out the back, and consider their safety in some way?UPDATE: I guess this should be a non-issue. Before someone rips the cabinet out, they'll have gone inside to disconnect the faucet. They'll certainly know about the electricals by that point.
 Subject to distance restrictions