Attempting to put caulking to the wall under the base board with the carpet installed is very likely going to result in caulking spread out onto the carpet - also if you ever have to pull it temporarily - the parts embedded into the caulk will separate from the rest of the carpet. Unfortunately, placing the caulking under the baseboard with the carpet removed also presents a problem; namely dirty, sharp, and possibly rusty pointy carpet grabbing spike heads - ready to puncture your fingers. I would recommend using caulking backer rod. This is a flexible solid foam rope which could be pushed into the gap conforming to the space and sealing off any air penetration.
A more permanent solution would require crawling;
If this is an interior wall and access to crawlspace or basement is available I would suggest inspecting the floor for penetrations which could be sealed. If it is an exterior wall check for penetrations through the wall plate (bottom and top) at the perimeter and any exterior penetrations for outlets. horizontal siding is notorious for allowing air to enter walls; without a well sealed vapor barrier under the siding, there is little that can be done without opening the walls.
If you can't close the door, it does not function so well either. Unplug the cooler, let it warm up completely to room temperature, maybe even add a little heat, but not too hot, warm to the touch, from a hair dryer, and give it a shove back in with the part of your hand that is near the wrist. You may need to pull it out of the space, lean it back a bit, but not too much, the refrigerant doesn't like it. and push it back in.
There was something that did that, it must have somehow been pushed severely from the back side, at least so I guess. Check the back for any sign of abuse that may keep it from going back in.
Before you try that, the crease is really predominate at the corner. After warming it up, put JUST your thumb over the middle of the sharpest part of the crease, and push steady and hard if you need to, the rest should follow. If you have a well rounded LARGE handle you may be able to use that instead. Since the handle has no feeling you will not be able to govern how the progress is going. It may be more prone to cracking the PVC shell that is deformed. If it does, humidity may enter the casing, that may or may not be ok.
Best Answer
The backing doesn't need to be very thick to prevent the deformation (parallelogramming) you are seeing because it works by tension rather than compression. The edges of heavier backing would show more and require molding to cover them. Lay the shelf unit down on its front, square it up and use small brads or fine finish nails to attach the thin backing around the edges--top, bottom and both sides.
Sagging shelves are another matter. Are these shelves adjustable, say with pegs in holes? If so, you would have to stiffen the shelves on the front.
If the shelves are fixed to the sides, you would tack the backing to the individual shelves, this would help, but the front edges of the shelves might need stiffeners attached to the bottoms along the front edge, if the shelves are not made of stiff wood. These stiffeners would interfere with getting books in and out and limit the size of books that could be shelved. You could put stiffeners on one or two shelves and put the heavier books there.