Is it valid for a person to embrace Islam between himself and his Lord, without declaring it, for fear of losing inheritance or suffering harm from his siblings? Is this inheritance considered unlawful? And is it permissible to take a loan from a bank to buy an apartment and rent it out, while paying a bribe to facilitate the process?

There is no objection to delaying the official documentation of one's conversion to Islam for a valid reason. Therefore, there is no sin upon the man for delaying the public declaration of his Islam for fear of his brothers.

Regarding the issue of his inheritance from his Christian father: If the father dies before the son embraces Islam, the son's right to inheritance is established. If the father dies after the son embraces Islam, the majority of scholars hold that a Muslim does not inherit from a non-believer. However, others permit a Muslim to inherit from a non-belligerent non-believer. The father's trade in alcohol does not deprive the son of inheritance, because alcohol is considered valuable property among the People of the Book (Ahl al-Dhimma).

As for borrowing money with a stipulated increase, it is forbidden usury (riba) and a grave sin.