Is the change of the tenant's name from the deceased father to the three children, and the payment of rent and expenses by the deceased mother, then the equal division of the rent among the children after her death, Islamically sound? And is the apartment in this case considered a gift? And should the proceeds from its sale be divided equally among the siblings?

We do not understand how to reconcile the apartment being rented with your question about distributing money equally in the event of its sale, as a tenant does not own what he rents through long-term possession. If the intention is for the tenant to continue benefiting from the property for a long time and to pay what is called "al-ataba" or "al-khuloo" (key money/premium) in exchange for vacating, then this is an injustice, and it is not permissible for the tenant to sell the property or take key money without the owner's consent.