Drywall – Must I replace drywall after very small fire

I'm assuming here that they're not going to replace the ducts in the wall; if I'm wrong I'll adjust my answer accordingly.

• Cut three or four pieces of 2x4's long enough to fit between the studs at the sides of the hole.
• Toe nail or screw one in horizontally near the top of the hole, one at the bottom and the other one or two in the middle. Provided your drill will fit in the space, screwing may be easier since you don't have to swing a hammer in a confined space. If you use nails, it would be a good idea to drill some pilot holes first. Use at least two at each side, one towards the front and one towards the back; ideally you would have them on the top and bottom of each cross piece too.
• Cut a piece of drywall to size -- it doesn't have to be exact since drywall tape is 2" wide -- but should be fairly close. If you have a drywall rasp, cut it a little larger than you need since it'll be easy to trim it down.
• Place it in the hole and screw it into the cross pieces.
• Apply some mud around the edges with the taping knife. Press the tape into it, then more mud on top, and smooth it out.
• Let the mud dry; sand it smoothish, then apply some more mud on top, spreading it out further. Repeat until it looks OK.

Tools:

• electric drill
• hammer (if you use nails)
• saw to cut the 2x4's (mitre saw is quickest, handsaw would do),
• drywall tools: craft knife, mud pan, taping knife, option drywall rasp (quicker to trim the piece of drywall down to size than using a knife), sander. (A few years ago, I bought a combination toolkit at Lowe's with all the tools you'd need this kind of job; it was a lot cheaper than buying them all individually).

Supplies:

• drywall
• drywall screws
• 3" (approx.) nails or wood screws,
• one or two 2x4's,
• mud (there are mixes that you can get that will dry in 30 minutes or so, but I'd recommend getting a bucket of all-purpose joint compound since this is your first time).
• drywall tape (I prefer paper).
• sandpaper (for the drywall sander)
• plastic sheeting to cover the nearby area (sanding drywall makes a lot of dust that you don't want in your vacuum).

None of the jobs takes particularly long, but you need to let the mud dry between coating and sanding, so figure a few days to get the wall ready for its finish treatment: texturing, painting, etc.

Any common construction or project adhesive. That said, drywall nails are fairly thin for just that reason. If it's large enough that you're installing a patch, a nail won't destroy the patch.

And that said, it's often the easiest solution to simply enlarge the patch to something suitable for repair, say 6" x 6". Float some one-by lumber or plywood scraps for backing and be happy.