What is this old 1/16" thick floor-covering, and how do I remove it?
Unless you have your heart set on using the wood floor under the old sheet goods I would cover it. As you suggest it may contain asbestos. I would use floor leveling compound, available at home centers to fill in the low spots where the old surface was removed. Then you can cover the entire floor with what ever material is suitable underlayment for your woodfloor be it floating,prefinished or solid wood.