Electrical – How to enclose wiring connections between a wall and a cabinet

You've got a lot of different questions, so I'll try to answer the ones I can:

1. Coax sucks because of the necessary bend. You're better off going with a backless box like the one you linked to, because it makes running cables so much easier. (if you have a second person, have them push the cables up from below (or a straightened coat hanger or other wire to act as a needle, if the bundle of cables is too floppy), and then snag it from the hole with a wire bent into a hook) ... which you can't do as easily if you mount a box with little punchouts. Also, you then don't have to worry about crowding the box, etc.

2. It's easier to run lots of cables at once. What what you're planning already, I'd likely run an extra cat5e/cat6 cable, and then get an HDMI-over-twisted-pair (aka HDMI-over-cat5) extender if you wanted to go that route, or re-terminate it for RJ11 if you wanted to change it over to phone.

3. I don't know of any other way to get cables/pipes through the sill other than drilling, and it's allowed for plumbing, electrical, etc, so long as you then spray foam the holes afterwards (at least, it's required in my area, might not be in yours).

4. You only need to line up well enough to get it in the same bay, so you have a ~14.5" target. I cut the hole for the box first, then check to verify the location of the studs (I don't use hardware that requires me to attach to a stud, so I just have to make sure to miss it when cutting my hole), then measure off of some reference point that I'll be able to find down below. (electrical runs, walls, etc.) If nothing else, you can use plumbing or electrical for identifying where the walls are from below.

5. To avoid humidity issues, use spray foam to seal around the cables. (see #3). It may even be required by code in your area.

You have two questions. To the first question of how to get the wire out of the existing box, you would mount a surface-mount box on top of the existing box and put a blank face place on it. Then the plastic wire mold could come out of the raised, blank box and run to the new raised box. The wire must always travel inside an approved box or conduit. You can't just have wire coming out of a hole as you have described.

For the number of wires that you need to run, the current National Electric Code (NEC) guidelines say that all switch boxes need to contain a neutral. If the existing switch box contains a neutral I would use a "4-core cable" which is sold as "14/3 w/ground" (14 gauge, three conductors) or "12/3 w/ground" (for 12 gauge - not common for lighting) to run to the new switch. If the existing box doesn't have a neutral, I'm not sure what code and local rules would say. I would think with such a minor change, you could just extend the wires as they currently are, and you are right that the white wire would need to be marked with black tape for the switch loop.

However, even if that is the "obvious thing to do" or "what most people would do", it might not be legal or proper, so you'll need to get clarification on that. If you have a neutral, extend it to the new switch and you'll be ok.