Drywall – Garage Fire Wall

Unless you have scrap laying around, you'll be buying a 4x8 sheet of drywall anyway (unless you can find damaged panels or the hardware store sells half sheets). So anything less than 4x8 will be about the same amount of work to patch, so cut out as much as you need. Just remember to try and end in the middle of a stud.

If it's 8ft. from floor to ceiling, cutting out all the way to the next stud would mean you'd only have to tape one seam. This would likely be easier than patching a bunch of holes.

I'm not 100% clear as to whether you bought the house with the cracks and the popped out nails, or whether it has happened since you bought the place. If it has happened since you bought the place then I would be concerned and have it looked into. If it was like that when you bought it the damage could have been there for decades, there's no way to know for sure unless you can speak to the former owner.

I'm assuming you've done this or you have a slab foundation, but if you have access to the foundation like a crawlspace or a basement have a look to see if there's any evidence of settling. Try and mark off the cracks and the positions of the nails in ways you can later go back and check for movement. Use a ruler to draw a pencil line perpendicular to the cracks every inch or so, and use a grease pencil for the nails, or a permanent marker. Keep an eye on them every few weeks and see what happens.