What is the ruling on the action of a wife who insists on attending weddings where there are forbidden things, such as intermingling, dancing, and the uncovering of 'awrah (parts of the body that should be covered), while taking the children with her, despite her husband's refusal and warning? And what is the duty to be done before and after her departure?

Your action of not attending weddings that include forbidden acts is correct and obligatory, and you must remain steadfast upon it. As for your wife, continue to advise her and remind her of Allah and His punishment, and explain to her that whoever gives up something for the sake of Allah, Allah will compensate him with something better. Warn her against Allah's wrath and a bad end. She should prioritize Allah's pleasure over her own desires, for therein lies the happiness of this world and the Hereafter, as Allah the Almighty said: ﴿But as for he who feared the position of his Lord and prevented the soul from [unlawful] inclination, Then indeed, Paradise will be [his] refuge.﴾ If she responds, then praise be to Allah. If she refuses, then use severity, rebuke, and boycotting with her; perhaps she will deter. If she attends, remind her of sincere repentance and the ugliness of what she did, and explain to her that a Muslim's life must be entirely devoted to obedience to Allah the Almighty, as Allah the Almighty said: ﴿Say, "Indeed, my prayer, my rites of sacrifice, my living and my dying are for Allah, Lord of the worlds."﴾ And that it is obligatory to adhere to Allah's law, as Allah the Almighty said: ﴿But no, by your Lord, they will not believe until they make you judge concerning that over which they dispute among themselves and then find within themselves no discomfort from what you have judged and submit in [full, willing] submission.﴾