yes
Any thinset will workb
You can do anything you want, this is America! but seriously, directly to drywall is a bad idea. Drywall will wick the moisture, and you will not get a good bond, and, could mold and mildew. use a 1/4" Backer board, like hardiboard, and screw that to the existing sheetrock and into the studs. Then, you can either use a thinset mortar on the backer board, and the cultured stone pieces, or construction adhesive. If you use like, liquid nails, you will want to get collored grout or caulking, prefereably grout, and fill in your joints. Stagger any cuts so that they do not all occur in the same area. Piece of cake.
same thing
With thinset.
White thinset
Modified thinset or An approved tile mortar.
There is no way to dissolve it. It must be physically scraped off.
The videos which suggest lifting a tile to check the thinset coverage on the back are right. If you are installing tile for the first time, this is a necessary step to be sure you have the technique right. For floor tile up to about 12" x 12", you'll use a 1/4" x 1/4" square-notched trowel. For larger tiles, you should use a bigger trowel - about 3/8" or so. Dip some thinset from the bucket, drop it on the floor, and then spread it out. Hold your trowel nearly vertical at the edge, with just a slight angle so you can drag out the thinset. It should leave deep grooves in the thinset that stand up on their own, and you should be able to see the floor surface between the grooves. When you place the tile, hold it as level as you can and then carefully place it on top of the thinset. Wiggle it back and forth slightly, as you apply LIGHT pressure downwards, and evenly. You want the thinset to make good contact on the back of your tile, but not so much that you flatten it out. If thinset is oozing up between tiles, you're pressing too hard. Check to see that there is no "lippage" - or uneven heights between adjoining tiles. The surface should be perfectly level to surrounding tiles. Peel up a tile, and check the thinset coverage on the back. It should cover at least 75% of the back of the tile. If it is less, then you're either not pressing hard enough, not using enough thinset or spreading it too thin, or your floor surface is too uneven. If all looks good, scrape the thinset off the tile and that section of floor where it was removed, and reapply and reinstall.
This sounds as though there is not enough thinset under the tile. Or your thinset didn't properly bond to the tile. There is no real way to fix this besides pulling the tile up. If they haven't already, the tile will likely start cracking.
Thinset tile cement.
it could be a number of things, premixed thinset, to much flex in the substrate, mixed thinset too thin, didnt seal grout "if in shower" where is the tile installed, what did you use to install it and how did you install it?
Mastic come pre mixed which thinset come in a bag that you have to mix with a mixer mastic is not good for your wet areas meaning showers or on the floor. Used mainly for back splashesr thinset is port base like cemet very strong used in wet areas outside for all types of tile installation