Drywall – How to keep leftover drywall mud from drying out in the box

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.

You certainly can put different types of mud on top of each other without any problems. Generally it's done the opposite way though - using the quick drying mud for the first/second coats and the pre-mix for the final coats.

As for sandability, pre-mix is going to be the easiest to sand. The quick drying mud can still be sanded without much trouble though, depending on the type you use. Typically you'll see 90 minute, 45 minute & 20 minute varieties. The faster drying ones sand harder than the slower drying ones. The 90 minute stuff is still very sandable, 45 is still doable, but I wouldn't want to sand the 20 minute stuff.

Also, keep in mind that the fast drying mud has a limited work time, and you can't just add more water to it to soften it up like you can with the pre-mix. If you mix some up, and it starts to set, you just have to throw it away and mix a new batch. This usually isn't an issue with the first couple coats since you go through mud pretty quickly, but the last couple coats are much thinner so you aren't using nearly as much mud.

One other tip - I usually keep two buckets of water around when I'm using dry mix mud. One with clean water that I use to mix the mud, and another with gray water that I use to clean my tools between batches. If you use the gray water to mix your mud, the mud will set faster than if you use clean water. Works out nice if you are using 90 minute but want it to set faster than 90 minutes.