"The cat closed the fridge door." would be the most natural way to say it in English. 'the door to the fridge' is the next best. 'the door of the fridge' sounds a little less natural than 'the door to the fridge' - HOWEVER, it is still grammatically correct and a native speaker might say it, so I think Duolingo should allow it.
"Freezer" is "congelador". There are lots of different terms for "refrigerator", depending on what country you're in, including "nevera". You can find some others in this thread: http://www.spanishdict.com/answers/245734/whats-is-the-difference-between-la-nevera-and-el-frigorifico