"Там нет дома."
Translation:There is no house there.
Because там is referring to a specific place. Where is there no house? Там. There. There is no house there.
"There is" can be confusing when translating. If you say "There is the house", you're using "there" to point out a location. «дом там». If you say "There is a house", you're simply stating the existence of a house but not where it is. «есть дом». In this case "there is" works as a phrase meaning "there exists", so the word "there" is no longer applying to a specific location, just general existence. That's just how English works.
Home doesn't usually refer to the building itself (unless you're a realtor), so it doesn't make as much sense in this particular context.
Still it doesn't mean you can freely replace the two words. You can say "two pets are missing" instead of "two cats are missing" - but you cannot translate them the same way.
In another course I read that you don't use "there" twice in English in the same sentence. Can someone explain?
It's not ideal, but there's little other way to say it with this meaning. Do bear in mind that "There is" at the beginning does not refer to location the way the ending "there" does.
No one says "there" twice but it must be translated incorrectly in this program.