Is the new husband of the deceased's mother entitled to inherit anything from the apartment or shop left by the deceased father after the distribution of the inheritance? And what is the ruling on the mother selling her share and her son's share of the inheritance to buy another property and register it in the son's name?

We understand that your share and your mother's share of your father's inheritance are an apartment and a shop. Your mother's new husband has no right to your father's inheritance unless your mother dies while he is married to her, in which case he inherits one-quarter due to the presence of her children. There is no need to ask about a hypothetical matter that has not yet occurred. It is permissible to sell your shares and buy a property in your name if your mother has no children other than you, as this would be a gift from her, and justice among children must be observed in such gifts.