Is zakat obligatory on shop rentals, and what is its amount? Is it permissible not to pay it fully to the corrupt government and instead pay it to relatives? Is it permissible to swear falsely to evade paying zakat to the corrupt government?

Zakat is not obligatory on rented shops themselves, but rather on the money generated from their rent if it reaches the nisab (threshold) and a hawl (full lunar year) passes over it. As for paying zakat to a corrupt ruler, scholars have differed on this matter. The Hanafis and Malikis held that it is not permissible to pay it to him if it is possible to refrain, while the Shafi'is and Imam Ahmad were of the opinion that it is obligatory to pay it to him if he demands it. They based their view on hadiths and narrations that support this, such as the hadith of Anas ibn Malik (may Allah be pleased with him): "If you pay the zakat to my messenger, then you are absolved of it before Allah and His Messenger, and its reward is for you, and its sin is upon whoever alters it." Accordingly, if the ruler is a Muslim oppressor and demands zakat, then pay it to him, especially if evasion leads to another forbidden act.