Does Sharia approve of a situation in which the husband's mother interferes in the couple's life in this manner, by counting his money daily, taking from it as she wishes, and entering the couple's apartment at any time? And is this her legitimate right according to Sharia?

If the husband's mother curses her son's wife, she is a wrongdoer. The husband must wisely balance honoring his mother with not wronging his wife. He must treat his wife kindly and be good to her family. As for the husband's mother entering her son's apartment without permission, it is not permissible, as it involves seeing private areas and contradicts the wife's right to independent housing. She (the mother) may take from her son's money what she needs for herself, and if the husband gives her what she wants as long as it does not harm him, it is a form of kindness to her. The wife has no right to object as long as the husband provides for her adequately. The wife must help the husband in honoring his parents and maintaining ties of kinship, for there is good in this world and the hereafter, and it is part of good character that weighs heavily in the scales on the Day of Judgment, and increases the husband's love and respect for his wife. The questioner must honor her parents and maintain ties with them, even if they sever ties with her.