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screwdriver
no
Off white
I guess it would depend on how close it is to the heat. It is plastic and can melt under high heat conditions. If it isn't located too close to the heat source, then I would certainly use it as a tidy solution to your needs.
HOLE SIZE
If you cut a hole in your drywall, try to do on side (or top or bottom) of access box frame along the edge of a stud. Use a stud finder if you are not sure where the wall stud is. Find the approximate center of stud with the stud finder, then measure about 1.75" off center and drill a small hole to guage if you are just past the edge of the stud. Cut your hole in the dry wall using the apron of the box as a template. That way, you can drill two small holes in the frame apron (the part that goes in the wall) and secure it to the stud with screws. The screws will be hidden by the frame outer edge. You can also glue it to the drywall but that is a permanent install and will damage the drywall if it ever has to be removed.
YES.
hinge
There is a flange which projects about 1/2" thru the opening...around the perimeter of the opening. It could be cut down with a saber saw or dremel tool, I suppose, so that it would be flush with the back of your cabinet. Use a fine tooth blade if you try this.