What is the ruling on using governmental (surveying) devices outside official working hours for non-obligatory work, or for private work, with the permission of the direct supervisor, and does it matter that they operate on electrical charge?

An employee must safeguard trust, and public funds are highly inviolable. The Prophet, peace and blessings be upon him, warned against taking public money without right. Therefore, an employee is not permitted to use work tools except for what has been explicitly or customarily authorized. If your superior is authorized to give permission or if it is customary, then there is no harm. Otherwise, it is not permissible. You can consult the regulations of the workplace or ask the specialists if in doubt.