The only real concern about asbestos is when you sand it and it gets micro size and then inhaled. Just being on the slab is not a concern health wise. You say your slab "sweats" you may have issues with any type of flooring. Aside from tile. If this project is designed to spend as little as possible and just to cover the slab than i think your plan will be fine. Ideally if money and time was not an issue, you would want to test for asbestos. Clean the slab, diagnose the moisture issue, fix said issue(s) and then lay youre perferred flooring.

1. You can leave it alone...it's not going to hurt you. It won't be any kind of a hazard unless it's pulverized. 2. In some jurisdictions and circumstances removal of asbestos-containing floor tile is not an asbestos removal job and requires no permits or special disposal requirements. Check with your state and municipality, but remember federal regulations always trump state and local ones. 3.The price you got for removal and disposal is a good one. For about $2 per square foot the stuff will be up and out of your house and you won't have had to lift a finger. 4. As a homeowner you can remove the stuff yourself and you won't be violating regulations...read them. Disposal of the trash is a different question however, and that is why you would have to discover the sanitation regulations and requirements for your jurisdiction. Good luck.

The brown stuff is some kind of mastic or contact construction adhesive. In places it seems to be on top of black stuff, indicating the black stuff is older than the brown glue. Maybe there was a congoleum glued to the walls at one point, or a previous backsplash used black pitch mastic. There was such a thing used to adhere Vitrolite (solid-colored plate glass) panels. If it would set your mind at ease, get it tested. Casey

How old is the house? The older types of mastic were called black mastic because they used asphalt based cement and they added asbestos to it, if that's the case they should have removed that also, they have special chemicals and you need good ventilation when using them. As being sticky, I never ran across this material, so I wouldn't know if it remains sticky or not after the tile is removed... I vaguely remember a little bit about it because when I was taking the building inspectors course ages back I did some reading on it, but that's about it, I'm sure you can find some updated information online, try to google "black mastic" and see what comes up. Good luck