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.

How old is the house? Is it even old enough to have had asbestos insulation? First thing I'd do is call that friend back and ask him why he used the word. This could quite easily be fiberglass or cellulose. Asbestos went off the market in MOST applications in '78 so if your house was built after 1980 it's very unlikely it has asbestos. Those who are saying not to vacuum it with a regular vacuum are spot on. The dangerous particles are not the ones you can see, they are micron sized. Leave everything be for the time being. If there is an actual concern based on the type and age, you can have a sample tested. It would not cost very much - maybe $25-$50 if you send the sample to a lab directly. But that's getting ahead of the game. What kind of insulation is this? Blow-in, batts, etc.? A picture would help.

Photo of the issue will help.