How to relocate a GFCI receptace from a wall to a basin cabinet side?

Question

Apparently, NEC 210.52(D) allows a GFCI receptacle output to be

"installed on the side [...] of the basin cabinet" [1]

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:

1. 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.

2. 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?

3. 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.

4. 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?

5. 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.

[1] Subject to distance restrictions

Answer 1

NM run to cabinet:

Add a metal junction box 1 foot down, but perhaps surface mount it to the wall in the "window" cut out of the cabinet back. Use a standard NM cable clamp secured in a knockout on the back of the box for the incoming NM. Use a blank box cover. This gives you mechanical protection for the incoming wires (no exposed cable) as well as accessibility.

Wiring from back of cabinet to new GFCI location:

Use a pre-wired non-metallic liquidtight "cable whip" to connect the cabinet back box to your new metal inside-cabinet GFCI box. It's easy to cut to the desired length and the liquidtight end connectors work on standard knockouts. Again, you've got no unprotected cable and liquidtight even protects against liquids.
Just connect the conductors and ground in each of the two boxes and you should be good to go.

Re: Ground

The incoming NM and the cable whip should have ground wires. In the cabinet-back box, connect a grounding pigtail wire (short green 12 gauge), the ground from the NM, and the ground from the cable whip together with a wire nut. Attach the other end of the grounding pigtail wire to the green grounding screw inside the metal box. You can buy green grounding screws or premade grounding pigtails with green screw attached if you need them.

In the new GFCI box, use a wire nut to connect the whip's ground wire to 2 grounding pigtail wires. One grounding pigtail goes to the metal box's green grounding screw. The other grounding pigtail goes to the GFCI receptacle's green grounding screw.

There are also other ways to do grounding, such as with special green grounding wire nuts that allow one of the wires to continue out the end of the wire nut, or looping a bare ground wire under the green box screw and having it continue on, but I'd stick with the basics. Get a pack of 10 or 20 pre-made grounding pigtail wires and some good quality wire nuts (like 3M) for 2-3 12 gauge wires, practice a bit on scrap wire, and you'll have a nice professional installation.