Is it permissible for a man to swim with his wife in the sea or a valley in complete privacy, and what is the maximum amount of her body a wife is allowed to uncover, and is it permissible for her to swim in the well-known swimwear?

There is no harm in a man swimming with his wife in the sea or a valley, and the like, if it is away from the sight of people. It is preferable to avoid this unless it is in a private place for the couple. There is no awrah (private part that must be covered) between spouses, so it is permissible for a woman to wear a swimsuit.