No they don't charge to moor at the quay, dont forget I went Monday afternoon, so it will probably be different with moire people around at weekends. The water isnt that deep there at low tide, in fact when I was there older kids were swimming off the quay, and were able to just about stand in the middle of the river.
The trip up the river to Wareham is nice and tranquil, and relaxing. It takes approx 20 minutes each way going at max 4 ish knots of course.
There are is a pub right on the quay which also serves food, and snacks, where you can sit outside if you want and watch the world go by. There is also another pub across the parking area which you can also sit outside at, right next to the public loos. Wareham itself is just round the corner, with the usual shops. You can go another 1/2 mile up the river from the Quay until you get to the bridge. It was nice and quiet when I went there... well worth a trip, just for some peace and quiet.