Is the questioner considered a pimp (dayyuth)? And is he obligated to do more than advise and remind his mother, for the sake of Allah, when she engages in relationships with men via phone and Facebook, especially since his father is aware of this and does not intervene? And does his father share in the sin, or is the latter solely responsible?

The questioner is not a cuckold (dayyuth). He is required to advise his mother gently and politely, and to prevent evil with gentleness, while observing the rights of the mother. This is because he is responsible for this, even if the father has neglected his duty, due to the saying of the Messenger of Allah, peace and blessings be upon him: "Whoever among you sees an evil, let him change it with his hand; if he cannot, then with his tongue; if he cannot, then with his heart—and that is the weakest of faith." Changing evil may become incumbent upon the one who sees it if no one else knows about it, or if no one else can remove it.