Is anything, or an expiation, or a charity required of the questioner and her mother for taking shoes that were not theirs, after losing their own shoes during the Farewell Tawaf?

There is no sin in taking the shoes for yourself and your mother if they are from trivial lost property that the one who lost them would not ask about, because the finder of insignificant lost property may benefit from it, and is not obligated to announce it. Nor is the pilgrim obligated to offer a sacrifice (dam) for purchasing goods after the Farewell Tawaf.