Is the expiation for an oath obligatory upon the one sworn upon if he breaks it, and is it obligatory for the questioner to pay it if he was the cause of the breaking of the oath and the other party will not pay the expiation?

What you mentioned as attributed to Imam Ahmad is a weak and less favored narration within the madhhab. The madhhab—as well as the majority of scholars—holds that the expiation (kaffarah) is upon the one who swore the oath, not upon the one concerning whom the oath was sworn. This is the relied-upon position among the Hanbalis. Whoever says, "By Allah, so-and-so will certainly do such-and-such," or swears an oath upon someone present who then does not obey, the one who swore the oath has broken it and the expiation is upon him. Fulfilling an oath (ibrar al-qasam) is sunnah and recommended, but it is not obligatory to fulfill the request of the one who swore the oath. There is no sin upon the one who does not fulfill the oath, even if they miss out on the virtue. If you do not fulfill the oath, then inform the one who swore it that the expiation is upon him, not upon you.