What is the ruling on a wife's hair being uncovered and her shaking hands with her adopted paternal uncle, and the occurrence of a dispute between the spouses due to the wife's unwillingness to confirm that her grandmother breastfed him, with the husband fearing the sin of his wife being uncovered before a man who is not a mahram?

Adoption is forbidden by Islamic law, and the adopted person does not acquire the rulings of mahramiyyah (being an unmarriageable kin). Therefore, the adopter must repent and restore matters to their true state. Your wife is a stranger to this adopted man, and she must observe the hijab in front of him. Continue to advise her kindly, while exercising the qawamah (guardianship/authority) that Allah has granted you. If she does not respond, she is nashiz (recalcitrant), and you may seek the help of sensible family members for advice. If she persists in her stubbornness, consider divorcing her, and do not forget to supplicate for her righteousness.