What is the ruling on shaking hands with an elderly strange woman, and what is the ruling on shaking hands with her if there is a barrier?

It is absolutely impermissible to shake hands with non-mahram women, whether they are young or old, or whether the one shaking hands is young or old, due to the risk of fitna (temptation/strife) involved. It has been authentically reported from the Prophet, peace and blessings be upon him, that he said: "Indeed, I do not shake hands with women." And from Aisha, may Allah be pleased with her, she said: "The hand of the Messenger of Allah never touched the hand of any woman at all; he would only take their pledge of allegiance verbally." There is no difference between shaking hands with a barrier or without a barrier, due to the generality of the proofs and to block the means (to sin).