Is it permissible to utilize a commercial property that was bought with mixed funds (85% halal and 15% haram) and to pay its monthly rent on behalf of the father?

Your father's renting of the shop from the profits of his trade, which are mixed with haram (unlawful earnings), does not affect the validity of the lease contract. The owner of mixed wealth is not forbidden from being dealt with, and the unlawful money is tied to the liability of its earner, not to the specific asset itself. Therefore, there is no legal impediment to utilizing the shop and paying the rent.