Is the apartment that the father bought for his son and registered in the mother's name considered the son's property, or is it considered part of the inheritance, especially given that the father had permitted one of his relatives to reside in it before his death?

Your statement that the apartment is yours does not make it so, for a gift is not complete until possession is taken, whether through use or registration, and that did not happen during your father's lifetime. In addition, your claim that the apartment is yours is not accepted without evidence (two just witnesses, or one male witness and two female witnesses), and your mother's testimony for you is not valid. The apartment is an inheritance for all heirs.