Is it considered a shortcoming regarding one's parents for the wife to visit her husband's family before her own family on Eids and similar occasions, and is it a religious obligation to visit all relatives to congratulate them on Eid?

There is no objection to visiting the husband's parents before visiting one's own parents, provided that this does not displease them. It is not considered negligence or undutifulness. Maintaining kinship ties (silat al-rahim) is generally obligatory and is achieved through visits, communication, correspondence, greetings, and anything considered by custom to be a form of connection. There is no specific measure or particular method for it prescribed in Islamic law.