How do you know when to use an "A" or "The" in these situations? Why wouldn't "The heavy wardrobe" work?
It depends on the word order. The sentence would have to start with the шкаф to use 'the' - the wardrobe has been mentioned before and now it's saying where it is. In the given sentence, it's describing what stands by the wall - a wardrobe. This is the first time the existence of this wardrobe gets mentioned, hence 'a' not 'the.
I think "there is a heavy wardrobe against the wall" should be accepted. "There is a heavy wardrobe at the wall" isn't natural to say although it's understood what the speaker means. We talk about things being "against the wall", not "at the wall".
Why wardrobe is a main option here? Where it says that there clothes?
Maybe the cupboard is full of books or dishes.
Something has to be the main option. They usually choose the most likely version to be the suggested translation, but as long as other versions are accepted it doesn't really matter.