What is the ruling on disconnecting the internet from one's father and brother due to their use of it for disobeying Allah, and is this considered an act of disobedience (عقوق), and what is the ruling on inspecting their devices?

It is not permissible to lend something to someone who will use it for impermissible purposes. Therefore, as long as they are using the internet for impermissible purposes, you should not allow it. Your action of disconnecting the internet is correct, because it is your property and under your authority. You should be gentle in addressing your father and remind him of your fear for him concerning Allah's punishment, and offer to compensate him with some gifts. There is no blame on you for inspecting the device as long as they have given you permission to do so; what is forbidden is spying.