Electrical – How to switch an HDMI cable through and existing wall (hole)

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.

Couple problems I see. First is the textured ceiling. You'll never be able to open that up and patch it without being obvious, the texture is something that just isn't done these days. Second, the exterior wall should have insulation in it, so you'll end up fishing the line through insulation and the vapor barrier.

Instead, I'd suggest running the line around the floor if you can get it to a convenient wall anywhere in that room. Use some fasteners to keep the cable at the base of the wall to run it around the room: