Is the pain emanating from the dying person—while they deny feeling it—indicative of their suffering in another world, or does Satan play a role in it?

It appears that what your father experienced was the throes of death, which every living being must endure. When a person is on their deathbed, they witness signs of good if they are a believer, and signs of evil if they are a disbeliever. Angels may descend upon the dying believer with glad tidings and reassurance. As for the coming of Satan at the time of death, it is not necessary for everyone; rather, it occurs to some and not others. You must pray for your father for mercy and forgiveness.