What should a religiously committed son do when his mother forces him to violate Islamic law, such as by wearing short trousers, shaving his beard, and shaking hands with non-mahram women, and beats and curses him when he refuses?

Explain to your mother - gently and politely - the religious ruling on the prohibition of uncovering the awrah (private parts), shaving the beard, and shaking hands with non-mahram (definition) women, and that you wish to adhere to obedience to Allah. If explanation does not work, then do not obey her in matters of disobedience; for there is no obedience to a creature in disobedience to Allah. If she harms you with hitting and cursing, then speak to someone who can prevent her. If you cannot free yourself from her harm, then be patient, seek reward from Allah, and resort to Allah. Continue to honor her and keep her company with kindness.