I just had 3 cm quartz countertop installed onto a completely serviceable set of 1979 walnut motif cabinets over a 50 foot x 2.2 foot wide kitchen cabinent run. (The only thing missing compared to today's products is the concealed European hinges/springs - happily I convinced my wife that my old hinges, although a bit old-fashioned, are serviceable, Counter guys came and templated the job putting a joint mid-point at the new composite white sink placement and at one side of the stovetop. The templaters were then the installers a week later. After they prepositioned the 3 pieces of ca. 500 pounds quartz pieces they called my attention to the fact that my cabinents were not level and proposed some "on the fly" leveling with wedges along the entire 50 foot run. Because that effort seemed to me to be unruly amd imprecise, I told them just to minimize discontinuities at the sink and cooktop joints while I wasn't looking. The project turned out great with an almost imperceptable lack of parallelism at the vertical at the front sink joint (This avoided shimming all over the place to correct a 0.5" runout error over the 50 foot run which except to a 0.16" modestly imperfect joint; because it is quartz, one cannot tell much by looking, but the level tells the story with which I easily live with!

I am a fabricator. Corner joint is botched, seam is poor. The varying gap between the underside of the stone and the cabinets is normal but typically it's filled with caulk after shimming. A quality install would have the installers taping off both sides of the gap before applying the caulk so the resulting caulk bead is in the gap and not smeared all over the stone and cabinets. Many shops don't do the taping because it is time consuming. However, most will at least caulk the gap even if they only use the wet finger approach (no taping) to finishing the caulk joint. My suggestion is to tell them after 4 tries they can have their stone and you want all your money refunded. then go find a competent fabricator.

We were asked to put 3/4 plywood on top of cabinets extending to overhang by the quartz fabricaters we used and they did a great job. Overhang seems very strong without any extra support other than ir OSB board underneath.

Cabinets have to be level, plumb, and coplanar, in all 6 of the cube’s dimensions. There are no corcumstances thatvwhat you propose is acceptable. Hire a professional cabinet installer. You won't own enough clamps to even start this. If you feel your flooring will be an issue, its short sighted to not include replacing it in a full gut remodel.