[susan53]

Let's swim - possible, but even more likely, Let's have a swim.
But for the others, it depends on what you mean by "in the water". If you're already up to your waist in water (for instance if you're in the sea but have just been standing up and playing with a ball, or in a swimming pool but have been standing an the shallow end for ten minutes talking to a friend), expressions with "go" would make no sense - you don't have to "go" anywhere as you're already there. But if you were just walking along the beach with just your feet in the water, then "go" makes more sense, as you do have to walk further into the water before you can start swimming.