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.

Yes you can put it over linoleum.

I vote for floating, click, wood look, luxury vinyl plank. I especially like the style with knots and chatter marks. Lay it throughout the whole house, on the diagonal, with no thresh holds. That's what I did. I can clean the entire house in approximately five minutes with an electric leaf blower. Porcelain tile is more durable however it is more expensive to buy, more expensive to install, and more expensive to uninstall. Also, some tile is extremely slippery when wet whereas wood grain LVP is not slippery at all. Five years later I am happier than ever I chose grey driftwood luxury vinyl plank for a house with dogs, kids, beach sand, and lots of wet feet. If the house gets inundated from storm surge and ocean water gets inside, I can unclick the LVP, clean and dry the subfloor, and then reinstall the same LVP and it will look as good as new.

Vinyl products do NOT like anything soft/squishy underneath them. In fact they can separate or break their own edges if you have underpadding underneath. If the Pergo vinyl has an attached underlayment, then it is DEFINITELY NOT allowed to add anything else. This is NORMAL for vinyl. In fact it is a major drawback for all vinyl products. Underpadding is not allowed with 99.9% of vinyl floating/glue down/perimeter glue products. The installation instructions will tell you what is ALLOWED to be used. If it is listed then it is allowed. If you do NOT see it listed, then it is NOT allowed. It is an easy rule to follow. Continue with the rigid underlayment (like you did in the bathroom). The noise reduction from the attached underlayment is the only padding/accustic insulation you are going to get.