Oxalic acid is good for getting up stains that are deep set in the wood (having gotten past the varnish)... it supposedly also works really well on those dark/black stains left when cats pee on carpet and it leaks through to the hardwood.
http://alsnetbiz.com/homeimprovement/oxalic_acid.html
For shine, tung oil would probably work pretty well and is easy to apply (just pour it on a lint-free cloth and wipe it on), but I have never heard of using it on floors. I have used it on a couple of pieces of furniture I refinished and it worked great.
If you are renting, you should probably ask your landlord before you put anything on the floor.
If you mix polyurethane 50/50 with mineral spirits, you could apply a couple of coats easily with a paint roller or paint brush and that would give you a good bit of shine.
Be prepared to run your A/C pretty hard in order to take the humidity out of the air and get tung oil or the poly/mineral spirits to dry. Also run a couple of box fans to circulate the air. Opening windows may or may not help, depending on the humidity level of the outside air.