Kinda looks like vinyl sheet flooring. Either way you must take up the flooring and scrape it very good. If its tile I would rent an air hammer with a flat chisel. After the floor is clean check for dips and bumps ( level is not so important) and level it out with floor leveler. Floating floor is a simpler option but I won't touch the stuff and you will get a more possessional job with engineered glue down. Better resale also.

You can install directly over the subfloor. It must be reasonably flat and secured well to eliminate squeaks and looseness. If you want to install a plywood layer, 3/8" minimum thickness. Reasearch vapor retarders and decide if that is appropriate for you. Note that asphalt impregnated retarders, such as roofing felt, can lead to momentary snapping noises sometimes...the wood sticks to the underside of the flooring and then releases resulting in the snapping noise. To avoid that use a slip sheet of kraft paper or choose a vapor retarder that has a kraft paper face.

Do some excavation to see what is actually under the top layer--you may find several layers of flooring such as vinyl, linoleum, plywood or even hardwood if you are lucky. Once you get to the last layer, you may find your floors level. It is odd that the only subfloor is in that room--mine don't have any at all, just the very thick tongue-and groove flooring itself laid directly on the joists. I need to do my own excavating in the kitchen--I know there are vinyl tiles (mine), 70s sheet vinyl (there when I moved in) and under that, who knows? From the basement I can see the same t and g boards as in the rest of the downstairs, so I'm hoping it is oak to match the rest of the first floor. My ideal find would be original linoleum in a nice pattern I could use! Whether you could install new flooring as you suggest depends on your joist spacing, the thickness of the flooring, and the weight of your cabinets and appliances. When I moved in, my kitchen had a stove in front of the chimney breast, a refrigerator blocking the only window, and 70s plywoodish cabinets along that wall and forming an ell on the neighboring wall. There was no sag or deflection of the floor. My kitchen is about 12x12, so the span isn't too great, and there is bridging between my joists as in most 1900s houses. I redid my cabinets to cover the wall to the right of the window (found the original doors in the garage and built my own boxes), moved the fridge to the pantry, and replaced the stove with a newer one of the same size in the same spot. Yes, I should have done the floor research at that time, but I just wanted the 70s crap gone. :) That sheet vinyl was under the cabinets, so it was put in before they put the cheap ones in--but it wasn't close to original to the house. If you opt for huge fridges and restaurant size stoves, you may have a problem--but my appliances are standard sizes since I'm just an average person and not a professional chef wannabe. Food tastes just as good done on normal appliances--it is all in the ingredients, not the utensils.

He won't warrant it because you have made a material change by installing new flooring. Your only options will be to tolerate it or remove the new flooring and screw things down then replace the new flooring. Too bad you didn't check the subfloor before you installed the new flooring. Houses shrink and settle a lot during the first year or two. Your home is a good illustration of why you might want to wait to finish a basement or install a moveable basement ceiling. It certainly makes access easier to deal with flooring, plumbing and electrical issues.