Is the house considered the property of the mother or the father in the case that the father built it as a dowry for the mother, then bequeathed that the mother's dowry be paid as a perpetual charity upon its sale, and then both parents passed away and the father relinquished his inheritance to the daughters?

The house that your father built for your mother and registered in her name is considered a compensation for the dowry, not a gift. Therefore, its ownership was transferred to her by contract, even if she did not take possession of it. Your father's actions of renting out the house and residing in it were with your wife's permission.

Accordingly, this house is considered part of your mother's estate. Your father inherits one-fourth, the daughters inherit two-thirds, and the remainder goes to the 'asabah (agnatic relatives) (her brothers or paternal uncles or their sons). If there are no 'asabah, then the remainder is returned to the daughters only.

If your father passed away, the one-fourth he inherited from the house would be divided among his heirs; the daughters would receive two-thirds, and the remaining one-third would be divided among his full brothers and sisters, with the male receiving the share of two females.