Drywall – Gypsum and … Equals drywall mud

I don't have any experience with boxes, but have quite a bit with the 5-gallon buckets of compound. Without knowing much about how those boxes seal, I'd recommend going with a bucket- the buckets/lids seal air tight which is important for keeping the mud from drying out. The other thing I'd recommend- every time you close the box or bucket, put a piece of plastic sheeting over the top of the mud. This traps the moisture in against the mud, and prolongs its shelf life. The buckets come with a round piece of plastic sheeting on top of the mud- I just fold it in half (to keep any bits of mud stuck to the plastic from being exposed to air) then when I am done for the day, unfold it, and put it back on top of the mud in the bucket.

Personally, I would just build a rectangle out of 2x4s slightly smaller than the hole, lower it into the hole flush with the existing framing, then screw it to the existing framing. That will give you a nice surface to attach your new drywall to. Depending on the size of the hole, you might want a cross member in the box as well so your drywall isn't spanning too big of a hole unsupported.

As for cutting back the existing drywall - I wouldn't bother with that for a couple reasons:

First, you're likely to run into issues with the nails or screws that were used to originally attach the drywall to that framing. If you cut around those, then the old drywall isn't going to have a very secure connection to the framing unless it was glued. And if it was glued, then you have the problem of the glue residue getting in the way of the new piece you want to put on.

Second, the damaged areas of the drywall are well within the areas that you'll need to mud over anyways - figure on mudding out at least 6-12 inches on either side of the joint. So you're really not saving yourself any work by cleaning that up.