What is the ruling of the Sharia regarding how to deal with a mother who insists on shaking hands with and kissing non-mahram men, and attending mixed-gender wedding parties, and what should the son do about this to avoid incurring her wrath and being considered a "dayyuth" (cuckold)?

Kissing non-mahram (definition) men is forbidden by the consensus of Muslims. Islamic law has strictly prohibited shaking hands with non-mahram women due to the corrupting influences that result from it. You must repeatedly advise your mother with gentleness and kindness. If she does not respond, seek help from someone who influences her. If she still does not respond after that, then you have fulfilled your duty. Your role then remains to accompany her with goodness and kindness and to supplicate for her frequently.