re-varnishing wood floor

I don't think you are going to be able to do anything that will look good without sanding what you have so it all looks the same. You may not need to go down to the wood, but only trying will tell.

Sure, you can just put a couple of coats of polyurethane, or you can do it the right way and strip the floor. Since it is a rental, why not talk to the landlord and either have him do something or at least agree to what you want to do.

And if you haven't, you best let your landlord know what you're planning _before_ you do anything other than clean/polish existing finish...

Nothing is liable to P-O a landlord than a renter doing some cobble-job to his property w/o his knowledge and/or approval. Could get you a bill to repair the whole floor professionally.

OTOH, if you know something about what you're actually doing, and talk to the landlord, he may be willing to provide materials if you're willing to do the labor (and take the consequences if screw it up).