What is the religious ruling on handshaking, what are the relevant proofs, what should be said to someone who asks about the reason for not shaking hands, and did the Rightly Guided Caliphs not shake hands?

The followed madhhabs (schools of thought) have agreed on the prohibition of shaking hands with non-mahram women (women with whom marriage is permissible), due to the saying of the Prophet, peace and blessings be upon him: "Indeed, I do not shake hands with women," and the saying of Aisha, may Allah be pleased with her: "The hand of the Messenger of Allah, peace and blessings be upon him, never touched the hand of a woman; he would only accept their allegiance verbally." The prohibition of shaking hands is among the limits set by Allah.