Is it permissible to go to the market or to charitable society gatherings for recreation and to purchase necessities, accompanied by one's mother and brother, with full adherence to modest dress, if the father objects to it?

The default is that a father feels compassion for his children and strives for their welfare. He is more farsighted and experienced, and obedience to him is obligatory in matters that do not involve disobedience to Allah. If you believe there is no justification for his strictness, then convince him or seek the help of someone who can convince him.

You should relieve yourselves in permissible ways, and you can seek the help of a brother to fulfill needs.

A married woman's husband has more authority over her than her parents. If he permits her to go out, her father has no right to object unless there is a legitimate Shariah-compliant reason.