"Since Paul is using the word house in 4,5, and 12 to mean family or household, it stands to reason it carries the same meaning in 15."
The word for family is close; its oikogennia.
"The word for family is close; its oikogennia."
I understand but the context of "house" in verses 4,5 and 12 is family or household otherwise Paul is telling Timothy to make sure he makes the beds and his clothes are picked up and make sure the building, the structure, obeys.
1 Tim. 3:4-5, "One that ruleth well his own house, having his children in subjection with all gravity; (For if a man know not how to rule his own house, how shall he take care of the church of God?)".
1 Tim. 3:12, "Let the deacons be the husbands of one wife, ruling their children and their own houses well".
The word that Paul uses for "house" and "houses" in these verses is oikos, the same word he uses in verse 15 for "house of God" and he says if Tim can't rule his own household how can he rule the assembly of God.