I'm with LWO: tear out the ceramic. It will be less trouble than pulling out/reinstalling the baseboards, potentially cutting the door down due to increased floor height, and boxing in the vanity, making it difficult to change out down the road. A clean slate would be my preferred option.

they need to come up before laying tile directly over them. if you think they may have asbestos, then send a small sample for testing. what i would do though is to open as many windows as you can, stick some fans inthe windows blowing out, and seal off the doors and vents with palstic. then using a wide scraper( i prefer the 12" with a long handle) scrape them up. i doubt very seriously tha tthey will flake much. if the tiles come up whole there is almost zero chance of stirring up asbestos. you could get a disposable suit and mask(not the paper ones get a decent mask) at any home improvement store in the paint area. if it makes you more comfortable, wet the area as you scrape to eliminate dust.

It will NOT look stupid. There are vinyl plank floors that imitate the real thing so well, that you have to get down on your hands and knees to discover it's not the real thing. You want to use a cementious skim coat floor filler to make for a smooth and flush substrate. Most of these have the binder incorporated in the powdered formulation, so there is no need to add a binder or a primer with any of those. But...read the product labels to know what you're buying and what performance you can expect. Of course, if the fillers available to you suggest adding a liquid binder or using a primer, then do that. A trowellable skim coat floor filler is done with a smooth edge trowel in several passes. After the first coat is set enough to walk on, it should be ready for another coat. Remember, a smooth and flush result is what you want to achieve. After the floor is absolutely smooth, then install your vinyl plank using the recommended adhesive.

No. Get the sledgehammer and create a proper base for the new flooring. If the rest of your flooring is a hardwood, that should BE your flooring in the kitchen as well. What is the flooring in adjacent spaces ?