Drywall – How should nail holes be repaired in a gypsum firewall

Since this is the accepted answer, I can't delete it. But after another decade of DIY work, I agree with the comments below. This is not code, and not safe. Device power cables are not intended to be run in walls and are a fire hazard. IMO, @gregmac's answer below is correct, provided you use proper electrical cables in the wall (as gregmac notes in the comments).

@Jeremy has a good solution, but if you're willing to do a little more work, there is a little better-looking answer. Since you're running an HDMI cable, I assume you have some kind of small stand/table in front of the outlet that your cable box sits in or on.

With a drywall knife, punch out a hole behind the TV and install a plastic "old-work" 1-gang box. Don't worry, we're not going to install electrics here; just a plastic box that has a couple of "ears" on it so it can slide into existing drywall. Do the same thing at the bottom of the wall, above your electrical outlet. You can now run the cables easily through the wall without patching any drywall. If everything's laid out correctly, both will be hidden (one by the TV, one by the stand/table that holds your cable box).

I would clean out the area around the holes to give myself room to work without being showered with cellulose. Open the holes a little: make them square, and big enough to be able to get your hands through. Next, get myself a roll of R-13 fiberglass batting (currently less than $10 at Lowe's or HD), and cut patches to fill the void in the cellulose insulation: you'd probably need two or three layers, depending on what size your ceiling joists are. Stuff them through the holes and spread them out. Finally, use a couple of 1x2's to cross each of the holes, and patch them as described in this answer to this question.

I've used a shop-vac for cleaning up blown cellulose, but for disposal, not for later re-installation. It seems to me that it has a very different texture before and after: clumpy and with some structure, versus powdery, no texture.