You might also find a friend who speaks Spanish to help translate or write things down and use Google Translate. It will be a start to communication.
There are other places to get fridge parts. I went online and found the door handle to our fridge for not a lot of money. If you ask, maybe he will be OK with taking it off the rent or maybe he'll get them himself. He may also have been told this fridge has everything and brought it right over...and found out things were not with it/missing.
In the meantime, I'd just breathe. You have something cold to put food in, yes? And it keeps things cold, yes? So stack the best you can and see what can be done about the shelves. If the worst thing you have to worry about is shelving, you're doing ok.
We are landlords. We just have one place and rent it to family. When the dishwasher died, DH had to take off work to deal with it and it was not fun. I realize you're busy, but so is your landlord and he probably thought it was still an OK fridge. Our fridge got slightly warmer then slightly warmer and it would "run" but not cool off. Weird things happen quite suddenly with appliances.
So, anyway, you said it took a long time to find the place and you just moved in and it sounds like a private rental. Give him a little bit of a break for the shelves.