I think "в её доме" is also an adverb phrase. You're right though - it's also a prepositional phrase because like the English version of "дома" it begins with a preposition. Prepositional phrase is always either a noun, adjective or adverb phrase as well. I think the "wrong" answers given should be accepted, but wonder whether the variety of answers that get across similar meanings are too complex for the algorithms to cope with?????
At least what I have observed to this point in the course is that additional information about the object is included in the "у ххх есть/нет" construction. Think of exercises where we have had to use уже, сейчас or ничего. This may be one of those "neutral" word orders the course favors and I cannot speak to ther possible translations. I do have to say that I think I'm getting the hang of using this structure. I got the sentence structure right, but I'm reporting anyway as I used desk instead of table for стол.
In English you need to use an article with regular nouns, but Mom is kind of an exception since it's also used almost as a proper noun (such as a name) in sentences. You could definitely say "The mom" if you're talking about any old mother, or just "Mom" if you're referring to your own mom.
@nir_sh - If you want to use the preposition в, it takes prepositional case, which makes дом -> доме (в доме).
If you were to say that, it would be putting a bigger emphasis on the fact that she has a chair inside of her house. It would almost be answering the question, "Where does mom have her chair?"
That is instead of the general idea of having a chair at home, like in this sentence, which sounds like the response to "Let's buy mom a chair!" (and then... "Mom has a chair at home", she doesn't need another one).