Does the husband's father have the right to order the wife to leave the house without her husband's permission to attend to a matter for one of his family members, even if the husband is aware, resentful, and silent out of deference to his father, and are there any Shar'i (Islamic legal) proofs the husband can present to his father to convince him to desist from such actions?

A husband's eagerness to avoid displeasing his father is a good thing. A righteous wife is one who helps her husband honor his parents and strive for the Hereafter, as stated in the Hadith: "The best [among you] is one with a tongue that remembers [Allah], a heart that is grateful, and a believing wife who assists him in his faith." The husband's father has no authority over his son's wife, and she is not obligated to obey him. Nevertheless, it is advisable not to refuse the father-in-law's requests if they cause no harm and if the husband permits them, as this helps the husband in honoring his father.