"Вэтомдоменичегонет."

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.

With nothing in the structure "house", it is almost impossible for the structure to be a "home", because a house (structure) has to be a place you can live, has to have something in it to enable you to dwell in it - although there are exceptional circumstances where people call places "home" which offer nothing or next to nothing to support dwelling there.

Кто-нибудь домой?

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)

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".