Construction adhesive may hold, but it also may release over time due to moisture exposure at floor level. You might want to consider a high grade silicone adhesive, instead. Regardless of what you use, you'll still end up staining the grout (a consideration only if you ever intend to take the island away). I believe that granite counter installers routinely depend on silicone and the weight of the granite to keep countertops in place. If you try adhesive and it fails at some future point, then I would mount a hidden frame of 2X4's to the floor and attach the cabinets to that. I would do this by drilling through the tiles. A hammer drill and the right bit will make short work of it. Then you can fasten the frame down with the something like tapcon screws. You could even add some silicone adhesive: this glue and screw combo should be plenty strong. As for powder actuated nail drivers, they take some skill to use correctly and get the depth correct, and are very dangerous to use without safety training: they're basically guns that fire nails instead of bullets. I'd discourage this approach.

You should not need to poly your paint in order for it to be "protected". The proper prep job will determine if you have issues with it chipping, and water spots shouldn't occur on fully cured paint. If the prep didn't include TSP, sanding, priming with an alkyd primer, and two top coats, then you will have issues with the finish regardless of if you apply poly. In fact, applying poly could accelerate the issues. Wait a week for the paint to be fully cured, and it should withstand normal wear just fine if you used a quality paint.

Thank you. Here are my vendors: Cabinets—Bilotta in Mt Kisco, Danielle Florie Brand is NAC, color: Decorator’s white Floor is porcelain from Terra Tile, Gina Counter—Secolo porcelain from Walker Zanger, fabricated by Ecuastone. Pendants—Pottery Barn

Step 1: Cut the individual colors out of the sheet they are in so you can see the color directly against the stone. White borders are useful when viewing a sample on a color you are going to be painting over, but not useful when comparing a color to an element that is staying. Step 2: Throw out any samples that clearly don't work with the stone Step 3: View the samples at different times of day and under daylight and artificial light. Throw out any samples that clearly don't work at each viewing. Step 4: Put the samples on the floor. Do they work with your floor? Throw out any that don't work. Step 5: Move the samples around the room. View them on vertical surfaces, not just flat on your counter. Throw out some more. Step 5: If you are keeping wall and trim color, test the remaining samples against those and throw out any remaining ones that don't work. If there are any other elements you are concerned about playing well with the cabinets, do the same for them. Step 6: Now with a much smaller pool of candidates, order large swatches (or sample paint so you can paint your own). Repeat steps 2 through 5 with the large samples. Step 7: Pick your favorite. If you can't pick a favorite by this time, they are all fine; throw a dart and don't stress.