A few additions to Jeff's advice if you decide to replace the damaged drywall.
I'd suggest you look into waterproof drywall or mold resistant drywall. There are various products that do a better job than the typical green board that you would use in the bathroom that will help if you are installing in an area that has a lot of exposure to the elements.
Double check the thickness of your drywall if you are patching. If you replace a 5/8" with 1/2", you'll notice it.
Use screws, especially on the ceiling. They take more time, but hold better and are less likely to cause the nail pops that you are seeing.
That being said, I think it's entirely possible that you're just more aware of drywall issues and there may not be anything wrong. To know for sure, I'd go up to the attic, pull up the insulation where you suspect damage, and look for water rings or other mold growth. Use a flashlight, no need to feel with your hands (and risk falling through). While you're there, check the insulation for blackening that would indicate mold is growing there, and if so, replace any damaged pieces.
Your ability to see the joints between the drywall is just a sign of not enough paint and/or primer. And popped nail heads are normal as a home moves. My personal plan is to replace all the popped nails with screws and double up on the spacing the next time I paint to minimize this issue.
From my point of view this metal part is a base plate which makes it possible to mount the fan onto the plastic parts.
As RedGrittyBrick noted, it is (or better was) zinc-plated. It seems to me, that most of the zinc layer is oxidized and cannot provide galvanic protection anymore. So why did it loose the zinc layer so quickly?
If it was behind a cover before, it might be, that the cover was to protect the fan and the base plate from humidity and corrosive vapours. Food can emit corrosive vapours like vinegar and almost certainly a lot of water even at temperatures below freezing point.
But the usage of bathroom cleaner might have been the death sentence for the zinc layer, because bathrom cleaners are highly acidic and can dissolve zinc quickly.
Anti rust primer can do the trick but you will have to coat the part carefully and without gaps. After that you can coat it with an additional protective layer. And you should try to get some stainless steel screws equivalent to those used there, as it is impossible to coat the screws.
There's one major drawback on using anti rust primer and other coatings inside your freezer. They tend to evaporate VOC and other nasty things in return which may enter your food. Something I wanted to avoid.
Alternatively you could clean the part thoroughly from any rust and screw some magnesium or zinc liner onto it. This liner will jump in for the lost zinc layer. And put the cover back on to reduce the amount of humidity reaching that part.
Best Answer
There are rusty metal primers available that will keep the rust from being any kind of problem. It is available in spray and brush on. Typically you will need to wire brush off any loose rust/material and paint it with that primer. After it dries the recommended time, it can be painted over with your color matched finish paint.