I think that's fine. I'd flag it as correct next time you see the sentence.
дом as home is a place where I live, my hearth, dwelling, shelter and hideout
дом as house is a physical building, a construction
For the latter, the words здание, строение and помещение can also be used depending on the situation.
This sentence is about the building.
Now I know what to say when I'm robbing a house with a Russian accomplice
Nothing like a double negative ! I ain't done nothing is idiomatic grammatically wrong English, but Russian loves the doubling up !
Anything is not a negative in the sense, if you used "nothing" then it would appear as double negative
I think he is referring to the Russian, as in Russian when you refer to nothing, never, nowhere, etc. you have to negate the verb (use a doubke negative). So in english we say "I never read." but in Russian we say "Я никогда не читаю." Which literally translates as "I never not read."
'There is nothing at this house' was not accepted, though it said that в can mean 'at'. Is something wrong with 'at this house' in this sentence?
Interesting question, because so far, I've only encountered people or animals "having" things, in the у [genitive pronoun] есть or у [genitive pronoun] нет formats. If things cannot "have" other things (у дома есть... or у дома нет... are invalid), then your suggested answer might be correct, because there isn't another way of saying that.
However, I think that "В этом дома" = "in this house" is key to translating this sentence, and has to be included in the answer, which precludes use of "This house has".
While the intent of your translation is more-or-less the same idea, it seems to me that it's too far away from the literal translation to be valid as an answer to this exercise. Artful translation might indulge you in the license you take with the wording, but it's not sufficiently accurate for a language course.
Yes, it is necessary. It's double negative, i think it's common in slavic languages (in Polish there is the same rule)
Yes, in my mind in English there is a slight difference between "house" and "home", but is that slight difference really so important as to mark a translation wrong?
How can a house be a home if there is nothing in it? Even homeless squatters brings things with them to make wherever they live "a home".
In English, you need "There is nothing" either before or after "in this house".