Is a husband dancing with his mother-in-law in front of an audience considered an insult to the wife's father and a violation of religious manners?

The wife's mother is a mahram, so it is permissible to shake hands with her, greet her, and be in seclusion with her, based on the Almighty's saying: "and your wives' mothers."

As for the husband dancing with his wife's mother in front of an audience, if the audience includes both men and women, then it is forbidden for the mother to dance in front of men, and it is forbidden for the husband to dance in front of women. This is among the reprehensible acts that necessitate repentance. Similarly, it is not permissible for your husband to dance with your mother, even if there are no non-mahram individuals present, because dancing is a gateway to fitna (temptation). The husband must repent and apologize, and the father must prevent his wife from dancing in front of men.