Normal cupboards are 34 1/2" from the floor. You can fudge it a little but if they are too short then a dishwasher won't fit under the counter and the stove will stick up too high. Another thing is where your drain comes out from the wall. It may be too high for you to lower your cabinets. Don't forget a garbage disposal hangs low and you need to hook a trap up.
I already noted 34-1/2"... Remember, I used to do this for a living. (Still do/semi-retired) So I am familiar and I have done a few. (Custom cabinets as well as stock installs).
That's the second time someone has mentioned a dishwasher... On a bottom cabinet box the bottom shelf is normally 3/4" thick, with the bottom at 4-3/4", the top of that at 5-1/2" and the top edge of the bottom rail lined up with the top edge of that shelf. That's just reference. So, under that bottom shelf, is the toe kick cutout... and dead space. So if you cut an inch out there, it would not affect the dimensions inside the box... for instance where a dishwasher might go... which we do not have a dishwasher, but who knows, someone down the line might want one, right?
The plumbing is fine and adjustments up and down are workable within specs. Your concerns on that were confusing though... After cutting down a cabinet, the concerns would be that the trap was too high, instead of low... But in my case is still not a worry.
Next, no worries on the stove and oven. They are already installed in the island I built, already installed. This I am talking about now is the wall/floor boxes.
Actually, putting the 3/4 inch ply inside the top edge would have been a great idea... If I hadn't built my boxes with a top frame. Years of doing these, I have this standing method of building a frame in the tops of 1"x4"... so that 3/4" is already there. Just easier for me doing installs to actually have some meat there to screw too (in the tops of boxes). Also stronger and keeps things square. I guess I could cut these out and replace with 3/4 ply, seal, then bond the stone directly to that...
Draw-backs to that method- First is that when you build boxes, the faces (rails and stiles) are planned to be wider than the boxes. You do this on purpose, so that if you have anything that is not perfectly inline, square or parallel, you have some room for adjustment. So from box to box, you would still have some voids between them.
The boxes are mounted to the wall and connect to each other at the faces. Plywood over the top then connects them all together again. The top is usually screwed from underneath through that ply... or with mine, if needed, straight through the top frame. If I do away with the ply, I'm trying to bond stone, not lay on a single top, right?
I don't know... I guess I could just cut off 1/2" -AND- screw on 1/4 ply. Glue (construction adhesive) and screw 1/4"hardy board... I was worrying that that might not be strong enough. But glued together would add "some" strength. Then, 1-1/4" stone and thinset is not silly putty. I'm thinking that would put the height at just below 36", now enough off the toe-kick that it is visually different from the island across from it and may be strong enough.
The plus would be that the bond would be stronger. The drawback to that is that I then would have to add a stone edge to the counter, instead of the planned hardwood wood trim to tie it all together. I would have nothing along the edge of that to nail to...