Refinish,  No stain unless you have to due discoloration, the floor guy can give you a read on that.   

Ask the inspector to take pix of discolored wood then post those pictures here.

Is the water discolored using the sink or the shower? 

There are special primers for the walls to encase the cat (or dog) smell from this--which is an option.With a floor like this, with about a million high heel divots from 1890 until yesterday, and a finish that has reached the end of its service life, the discoloration is not such a problem, if you intend to put in pet resistant carpet and padding, it will be covered.An alternative would be to sand it down and refinish it to look like a refinished floor...they never look like new because they use chemical clear coats now vs Varnish or shellac like the olden days.Even with sanding the dog stains may need to be bleached out, and may still have trace shadow that they were there.Why is it a computer nook, well it doesn't have to be.

Still holding up well in some of our properties.Our difficulty with the acrylic and plastic surrounds occurred when they discolored from tenants burning candles, by being really grimy, or just with age.

The walls have mild discoloration which I plan to wash and then subsequently paint by myself.

In the past when I have had to deal with smoke, and discolored walls, I would do one good scrubbing with TSP. 

How much does floor one want for the damage.A hole in the ceiling with Spackle, the splatter finish matched (possibly whole room ceiling re splattered), then a primer and paint; or just water discoloration remediation?

The tenant had put a plant pot on the hard wood floor on the first floor which caused damage (discoloration of the wood with an uneven surface).  

I use a 12 inch ruler in the photos next to damages such as carpet stains, repaired holes in doors if they are still visible, bent window blinds, discolored fireplace brick, etc.