Yes a mini fridge can be placed safely on carpet.
There's no danger in placing the mini fridge on the carpet and it will not get hot enough to cause any issues.
I have a mini fridge on the carpet in my bedroom and it's been there for 3 years now and it's never been an issue.
I've also had a mini fridge on carpet in a dorm room as well when I was in college and it never was an issue.
The compressor gets hot but not hot enough to cause any fires or anything.
The only thing that could happen is if the refrigerator shuts off and then leaks water from thawing out onto the carpet but that's not much of a problem anyway.