What are the rights and duties of the resident Imam towards the mosque administration and vice versa? And is it permissible for the Imam to oppose the administration in administrative matters that do not contravene the Sharia?

The reference point for the matter is the mosque's situation. If there is an administration tasked with managing the mosque, then the Imam must obey it in administrative matters that serve the common good and do not violate Islamic law. The administration, in turn, must respect the Imam and appreciate his knowledge, especially if he is a memorizer of the Qur'an, and pay his dues.