The black felt-like paper is probably roofing felt. It's commonly used as a vapor barrier between the 'room' below and the kitchen. If it were possible to lay in some felt where it will overlap EXISTING felt by, say, 3-4", I'd say do it. But if the felt is simply cut-to-fit, moisture will come through at the 'seams'. (Not what you want.) As for leveling the floor, there are floor leveling materials out there. BUT, you need to read the instructions to be sure it can be used in your particular application. There are other approaches to this problem, and a new floor is what I had in mind. Yes, it will cost much more than what you want to do but you have to evaluate both solutions.
If you want to install a new floor, you can save some $$$ by doing the demolition and disposal yourself. Just get 3 estimates for the demo and disp AND the install, and see if it's worth your time and $$$ to do the demo and disp yourself, or just worth paying the contractor. There's a few posts on the Forum about this particular project. One is Linoleum. Search the main Forum page for keywords like 'vinyl', 'linoleum' (sp), 'tile', and 'asbestos'. My best to ya and hope this helps.