Am I obligated to offer expiation for an oath (kaffarat al-yamin) and am I sinful if I do not give it on behalf of someone whom I caused to break their oath?

The expiation (kaffara) is related to the one who swears, not to the object of the oath. However, sin may be related to the object of the oath if breaking the oath involves sin or mischief. "Fulfilling an oath" (ibrar al-muqsim) is a Sunnah in matters that do not involve mischief or sin, based on the hadith of Al-Bara' bin 'Azib: "The Prophet, peace and blessings be upon him, commanded us to do seven things," and he mentioned among them: "...and fulfilling an oath."