I'd try to get a piece of 5/8". Usually they come in 2x 4x8' sheets, which is overkill. I'd consider trying to find a scrap piece either from someone else's remodel, or a Habitat Restore Store, or check for partially damaged pieces at Lowe's or HD. If you ask any of these, they'll likely let you have it for free or very cheap.
If the plumber cut on three sides and folded it out somehow, that piece should be able to be put back roughly flush, at least enough to where after you're done it shouldn't be noticeable. I wouldn't do what you're describing there just to have non-angled cuts - it's all going to be hidden, right? If it bothers you, you can, of course, but you're just making more work for yourself.
On the lower portion, I'd probably remove that section of baseboard (it'll likely be nailed in with finish nails and then caulked along the top where it meets the drywall, and in the corners. If you do it carefully, you can replace it when you're done. Then cut that strip of drywall on the left where it's covering the stud, and cut to the right until you find the next stud over (probably 18" on center). Then cut a scrap piece of 5/8" drywall (see above) to fit and screw it into the studs.
One thing I can't tell is what's going on with your toilet water supply line there where it goes into the baseboard. Someone more expert than me can advise you on that maybe, because it might complicate getting that piece of baseboard removed.
You'll want to use drywall corner tape if you have it in that left corner. If not, there may be kludges that you might find acceptable. In order to install the drywall corner tape, you may have to sand down the texture on the (yet unmarred) wall to the left. Once it's taped and mudded, then you can hopefully restore the texture, then reinstall baseboard, then re-caulk, then paint.
