Is the salary received by the imam of a mosque considered unlawful if he is late for some of the obligatory prayers in leading the worshippers due to certain circumstances, and his brother deputizes for him in these cases?

Whoever is appointed as an Imam for a mosque and is allocated a salary, it is obligatory upon him to fully commit to the Imamate and not to be absent except out of necessity or need. Otherwise, he would be sinful, and the money for the days he was absent without a valid excuse would be illicit and forbidden wealth. It must be returned to its rightful owners or spent on general Muslim interests. Appointing someone else does not absolve him of his responsibility nor does it make his salary permissible.