Is the youngest son entitled to claim the apartment's rent for the remaining period of the contract after his father's death, and is the usufructuary entitled to claim compensation for being deprived of using the annex during that period?

This issue is a matter of dispute and should be referred to the Sharia courts. Agency for a known wage is binding according to some scholars, and thus it is not revoked by the death of the principal. According to this opinion, the heir is obliged to complete the contract period, and the agent is entitled to his wage by offering himself and enabling the utilization of the benefit. Therefore, the heir has no right to demand the wage for the two years or to prevent him from utilizing the annex, as it is part of the wage. The questioner has the right to demand compensation for not using the annex during the last two years.