Given what looks like black mold on the lower portion of the other drywall I'd be tempted to cut a large rectangle. I'd probably opt for the paper free green board that can't mold. The sheet will be 4x8 so I'd pull back a 8' section along the bottom and cut it to the height of the current hole. Keep the drywall 1/4" off the concrete. Your plan for 2x4 ...
A little too perfect; it'll be protected by trim. It could just be a 2x4 screwed to the surface of the stud and one to the floor plate. And maybe a floating one in the top right. Cut the hole square first.
Your proposed solution is perfect. I'd suggest using pressure treated wood in contact with the concrete. Adhesive probably won't work terribly well, but if you get a couple of screws in on an angle on the right (connecting to the old plate) plus lots of screws elsewhere, everything should hold together well enough to carry drywall.
A strong magnet on a string! It works like magic on plaster and costs less than $5!
Get a small, strong magnet Be careful! Especially with neodymium magnets and children/pets and around other metal/magnets/credit cards!
Tie the magnet (or stack of magnets) to a thin string, roughly 1.5-2 feet long.
Slowly drag the magnet sideways across the wall, so that it'...