This answer is a general overview of the points you are asking about.
I do not know the size and scope of your project so i can not tailor my answer to your specific situation.
I think ideally, I could cut exactly down the center of a stud so that
I could screw it back in.
Yes, that is what i do, i am practiced and can do it well if i take the time to find the stud and determine if it is not plumb. You can use small drill bit to find each side of the stud so you can find center. I use an oscillating saw for this and it is great , you may encounter some screws or nails, just go around/over them. You are going to be covering this seam up with tape and mud. (Taping and muding a but joint requires the mud to be tapered/feathered out quite a way from the the seam, it is a skill that takes much practice so you will not notice it.)
It might be easier to cut along the edge of the stud, but then when
putting it back in, there's nothing to screw into, unless I add some
I do not want to take the time to find the stud and determine if it is not plumb then i cut on the edge of the stud with A Rotozip (rotary cutting tool) and then sister in another piece of wood to the stud for attaching the drywall to.
Last idea was to cut significantly in from the studs and just put it
back in place kind of like a hole, hoping the mud will hold it in
Mud alone is not sufficient for that for patches over a certain size (there are specific techniques for doing small patches that utilize the paper of the patch to aid in the strength of the patch).
If i cut a hole in place where there is no structure to support the patch piece i take some 1x4 and screw it to the backside of the drywall, i hold it in place so it is partially behind the drywall edge and partially exposed, then drive some screws through the drywall into the wood. Now the new piece can rest against it and i can screw the patch piece to it.