Get a flat-bladed screwdriver, rest the tip in one of the notches, and tap the handle gently with a hammer. Alternate occasionally between notches to reduce cumulative damage to he faucet stem. The shock should loosen the nut.

They come in a set.

If it happens once upon a midnight dreary, it could be a raven.

Install a T-Plate back under the countertop where the particle board is still sound. Bend the plate if necessary so that the furthest hole lines up to where the existing screw is supposed to go. Then mount your dishwasher to the T-plate.

Putting it in the sun is a good method. Resist the urge to chip at it with something hard - that will likely just damage the interior plastic.

Also, it’s possible that the fridge seal is not the problem. There should be a vent where cold air from the freezer enters the lower fridge area. Make sure that’s clear. Otherwise the cold air is just staying in the freezer, so you turn up the cooling to make your fridge cold and the freezer gets extra cold. If that’s what is happening then this is the time to rectify that.

Those panels are attached with finishing nails (see second photo). If you pry the panel away from the stud you may be able to get ahold of the nail head and pull them out. Then just slide the panel to the right or left until it’s no longer trapped behind pipes.

Note that you’ll still have to remove anything attached to the paneling, and if you can’t move the whole panel right or left you may need to cut it up. An oscillating cutter or even a dremel tool would be good to have.

I would screw eyebolts through the plywood near the corners, then use S-hooks to hook to the grill. If you’re concerned about the weight, add more eyebolts on either side of the shelf.