Fisrt, DON'T REPLACE THE FLOOR, that would lead to a lot of headaches and is probably un-nessassary. That floor shouldn't absorb oil that quickly if it had been sealed properly. My guess is that the floor either wasn't sealed correctly or it's been so long since it has been that it's all been worn off.
Try getting a wood oil soap (like Murphy's or Old English) these soaps have oils in them that are meant to be absorbed into wood. This won't take out the oil, but it might help to make it blend in. I don't think that stain is comming out, even if you sanded the floor, the stain would have absorbed into the wood too deeply.
It's important to remember that wood is pourous and the natural oils that are in it are evaporated over time and have to be replaced, or else the wood becomes discolored and brittle, which is probably the case here. I wash all my wood floors twice a year with an oil soap to help keep them hydrated.
You raise a number of issues.
How to fill an irregularly shaped hole?
To cut drywall (or any panel) to an exact shape with non parallel sides is hard. So what we usually do is make the hole regular. You can trim away some of the remaining drywall, preferably so the edge runs along the middle of a stud, parallel to another stud or the corner. Then cut a rectangular patch to fill the hole, leaving about 1/8 inch gap.
How to make a patch sit in the same plane as the wall?
In general, we want the whole wall to be on the same plane (when you place a straightedge on the face of one section, it sits flat on the adjacent section, neither gapped or raised). If a patch will sit below the level of adjacent wall, you can put shims behind the patch just thick enough to raise the surface to flush. You can use trim boards or tapered shims to get the right thickness, tack them in place with brads, and drill through them when installing the drywall. If the patch sits above the surface, its usually best to use thinner material (but don't violate fire laws about minimum thickness).
In your second picture, the patch looks like it is about 1/2+ lower than the wall. You should shim it out. Even though you could use tape or drywall corners to cover this zigzag, it's a weak, thin joint and will look odd.
How big a gap is acceptable?
Between panels, 1/4 inch is fine. Taping compound and tape will cover it with little difficulty. Putting on a coat of compound before setting the tape will fill the gap. If you are using mesh self adhesive tape, forcing a bit of compound through the tape into the gap couldn't hurt.
If you have a gap that is approaching 1/2 inch or more, you can run into a problem with shrinkage and strength in the joint. Consider recutting the patch. You could use a thin sliver of drywall to partially fill the gap before taping, but it will be a bit more of a challenge to get it level and smooth.
Best Answer
If it is a lot of oil you may need more than paint. I would go over the area with something like drylok and let it sit for a week or two. If nothing comes through then just repaint over the drylok. If you see any sort of oil coming through - even a tiny bit - then I would skimcoat the area with drywall mud. Then repaint after that. I have personally tried to repaint over oil stains totally wasted my time, noticing the same oil coming through a few days later on new paint job. I am sure this has to do with what kind of oil, how much, and the paints you use (oil based paint will hid this better than latex).