Is it permissible for my mother to perform the current obligatory prayers standing, and make up the missed prayers sitting, given her difficulty in standing? And is it permissible for her to make up the missed prayers in Ramadan with the intention of making them up, instead of performing Taraweeh sitting?

Making up intentionally missed prayers is a matter of scholarly disagreement, but making them up is more prudent. They should be made up by focusing only on the essential pillars (arkān) and obligatory acts (wājibāt). Standing is required if one is able, but if standing harms the worshipper or becomes difficult for her, she is permitted to pray sitting, based on the Prophet's (peace be upon him) saying to Imran ibn Husayn: "Pray standing; if you are unable, then sitting; and if you are unable, then on your side."