I’m not a pro so I am reluctant to advise on this, but here goes: Patching with drywall is probably the best choice. Drywall can be such a pain in the butt. So messy! I think that 1 inch gap may be too big for any type of paintable caulking, and the caulk would fall behind the wall, but others may be able to chime in on this. If you could find a piece of molding (small-thin) to add into that crack, with paintable caulk to make it blend in, it may work, but it not done “right” it could detract from the stone and look cheap. Also should paint it the same color and sheen as the walls to keep it from “popping”. If your stone sticks out some then you could get molding that is concaved (I don’t know what it’s called). This is all I can think of. I hope you are able to find a solution that doesn’t detract from your beautiful stone!

No, the legs support the weight. Alternatively you could put a cleat on the whole back wall and skip the back legs, using the front legs for fine adjustments. Obviously the cleat has to be perfectly level. If you use the cleat I would skip the metal hanging rail,

Do NOT fix laminate floor edges with filler. These are NOT wood floors. The click system is not engaged properly. Whomsoever installed the floors did not get the edges to click properly. To *fix the one plank with the gap in the middle of the plank, you MUST fix the planks around it. If you look above the plank at the two planks (short edges meeting) you can see the planks are NOT perfectly aligned. Even that 'snick' of a misalignment and you get the gap on ANOTHER RUN. In other words, fix the alignment for the planks AROUND the gapping. Or else this is going to get worse. The floor looks newly installed. How old is it? Who installed it?

Backer and caulk and then a piece of square moulding.