Does a passenger’s refusal to give up his seat to an unfamiliar woman on public transportation, after being asked to do so by an elder in order to improve the image of Islam, constitute an offense against Islam and Muslims, and is there a sin upon one who does not wish to stand for anyone?

Allah loves the doers of good and has mercy on the merciful. Goodness and mercy are مطلوب (desired/required), and this is particularly emphasized with the elderly Muslim, as the Prophet, peace and blessings be upon him, said: "Indeed, part of glorifying Allah is honoring the elderly Muslim." Furthermore, showing kindness to non-Muslims who are not at war is also desired if there is a legitimate benefit in it. What you are doing is a supererogatory act, not an obligation, and you have a greater right to your place which you reached first.