There were probably no clearly defined "house" structures after the geneologies, since there were only three major Dwarven settlements in the First Age (Belegost, Nogrod, and Khazad-dûm). Even though, I believe that one Ring went to the currently living descendent of each of the fathers. Thráin had one of the Seven, and was descendent from Dúrin. Also, a Ring was at the base of all the seven "hoardes" of the Dwarves (can't find the exact quote) and it makes sense that, having no other obvious divisions, the seven groups would each have one hoard.
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