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.

If you are worried, send a sample to a reputable lab to be tested. It runs $25 - $50 and you'll have the results in 1-3 days. Although asbestos backing on vinyl flooring was outlawed, some shady flooring retailers were still installing it after 1984. We found that out just before we removed flooring from our kitchen!

It's really a matter of how well the adhered tiles are stuck. If you can get something like one of those roofing tear up bars underneath and pop the tiles up in full pieces, you aren't likely to generate much asbestos dust. The adhesive can have asbestos also so it could be wet scraped and picked up with a wet vac with a capture bag. If the tiles are adhered such that they break into little pieces, you can likely float where the loose tiles came up and put a new floor over the top. There's also floating vinyl floors that sound like an ideal solution for your situation but I know little about the efficacy or quality of these products.

Airborne is just that, in the air. Anything that was in the air will have settled onto/into anything that is NOW in the space. That includes air vents, air ducts, exhaust fans, air return outlets, electrical sockets, etc., etc., etc. A professional mitigation company can come in a deal with the building envelope and the stuff attached to the building (air vents, HVAC, etc). Walls can be dealt with by the same company, etc. The soft furnishings are now "contaminated". Like any dust particle, the drapes, bedding, etc are now covered in that asbestos dust. They can be professionally cleaned (ask the remediation company who they would recommend). The bedding, clothing, etc can be tossed into washers/driers...that should do it....but again you want to be proactive about cleaning vents etc. Exposure is negligible - in other words I wouldn't worry about it re: a health scare. It takes quite a bit of time (think miner in an Asbestos mine without masks) to get sick from it. But the problem still remains: remediation was not done and asbestos was ?illegally? removed/disposed of. Dust now exists on every surface, in every air vent, in every nook and cranny. You are now aware of the existence of asbestos in the building envelope and will be legally required to properly remediate any/all surfaces that contain asbestos AND lead (possible if the earliest paint layers are pre-1978). I would look at this from a legal stand point...not necessarily a "health" stand point. Start with the remediation experts in your area. Follow their advice - to the letter. If this will become a rental building and not just a family home, consult a lawyer. Stop hubby from doing anything else on his own - for now. Testing is to be done FIRST...then he gets to swing hammer. The age of the building will tell you what the risks are....anything prior to 1978 had a HUGE risk of both asbestos (confirmed) and lead (paint and pipes). And just to be clear, the clean-up at this point could be more expensive simply because it now includes EVERYWHERE (every surface on all three stories). Had a remediation company come in and dealt with this single room for you, they could have contained the dust to this area....no need to worry about dust in the air vents and the air returns, etc throughout the building. Draping off a single room is easy/cheap. Cleaning up a 3 story building = expensive.