Is it permissible to pay an estimated Zakat on behalf of my deceased father who did not pay the full Zakat on his wealth, and is it permissible to give this Zakat to my needy mother and sister?

If you doubt whether your father paid the zakat on his wealth, the default is that nothing is obligatory upon you. However, if you are certain that he did not pay it, then it is obligatory to pay the zakat from his wealth before the division of the inheritance. If you do not know the exact amount, then make an effort to determine it until you are reasonably sure that his obligation has been fulfilled.

As for giving zakat to the wife and daughter, it is not permissible if their shares of the inheritance remove them from the description of being poor. It is permissible, however, if it is to settle a debt they owe. The jurists differed on giving them from their deceased relative's zakat due to poverty, but it is safer not to give it to them to avoid disagreement.