Just sanding it seems futile - you can't really sand gooey wet stain and expect a happy result.
1) Spend two days on your hands and knees with a razor blade in one hand and a rag soaked with mineral spirits in the other. Peel up all the places where the polyurethane moves independently of the floor (this can be tested by pushing on it with your thumb, or by using the rounded end of the razor blade), and then wipe up the gooey stuff with the mineral spirits.
2) Pull up all the carpet staples and sink any nails that need sinking.
3) Rent a drum sander from Home Despot. Get a lot of 36 grit sandpaper. Also get some 60 grit and some 100 grit. Sand the shit out of your floor. Think you might be able to fit this into the 4 hour rental window? Think again - there's a lot more goo that you weren't able to get up. Try not to dig a huge hole in your floor the first few times you start the thing up, or 5 hours later when your arms are jello. Be proud that you were able to get up all the cat piss damaged stuff that you've been told can't be fixed unless you replace the entire floor.
4) Go to Home Despot and rent an edger. Do the edges. Be careful not to cut too deeply into the area where you've already gotten down to the bare wood with the other sander, or you end up with circular marks that are not necessarily easy to get out later.
5) Discover that despite the name, the edger doesn't actually get ALL the way to the edges, and particularly not into corners. Follow up with an oscillating hand sander to even things out to the edges.
6) Realize that the there's cracks and holes in the wood that need filling. Yes, you should have done prior to step 3. Too bad. Fill the damn cracks now with wood putty and let it dry, then go around on your hands and knees with the hand sander and sand it down.
7) Clean thoroughly.
8) Repeat step 7 until you can't imagine that a speck of dust would dare raise it's ugly little head.
9) Stain. This is easiest with 2 people. One person applies the stain, the other follows behind to wiping up the excess and feathering the edges. It is best to follow the grain, and stain along with the wood strips. That way you reduce the number of places where you have visible changes in the stain. Let it dry at least 6 hours. Probably overnight.
10) Apply the first coat of polyurethane. If you use the water-based shit, you can apply the next coat in 3-6 hours without sanding between coats. But you should not apply more than 2 coats a day. After I hit "submit," I will go and apply the second coat for the day. Then tomorrow I'll sand, wash down with mineral spirits and let it dry, and then apply coats 3 and 4.
If all goes well, we'll be able to move back into our bedroom sometime next week.
Except, of course, for the bit that involves carrying furniture from the pod to the bedroom.