Yes, "Häuser" becomes "Häusern" in the dative plural. All nouns add an -(e)n in the dative plural. The only exception are nouns whose nominative plural already ends in -n (e.g. die Frauen) or ends in -s (e.g. die Autos).

Here, however, "houses" is the direct object. For almost all direct objects, you have to use the accusative case, not the dative. That's why no -n is added.

Yes. Thats how they taught it in school, and it really makes it clearer. I wish this app would use and accept "y'all" instead of inferring you (guys)

"Hast du Häuser?" is a question. Possible translations of "You have houses" are:

1) Du hast Häuser. [informal singular]

2) Ihr habt Häuser. [informal plural]

3) Sie haben Häuser. [formal singular or plural]

Yes, "du hat" is wrong. "Hat" is used with "er/sie/es" (he/she/it); "hast" is used with "du" (you [informal, singular]).

er/sie/es hat = he/she/it has

du hast = you have

Because that would mean "You have the houses" -- but that's not what the English sentence says. It doesn't talk about specific houses that you have spoken about previously ("the houses") but just about indefinite houses in general ("houses").