You've asked some good questions, but the answers are more complicated than the questions.
There are campsites throughout the canyon, and a four day (three night) trip would have most people camping in Pate Valley the first night (lots of campsites here) the second night below Muir Gorge or above or below Cathedral Creek, and the third night near Glen Aulin, and then out on the final day. There are campsites in all of those locations. And there are campsites along this whole route, with the exception of the rock secton of Muir Gorge, where the trail actually swings away from the river for a while to navigate through the granite.
That said, I am not sure what you mean by finding campsites far from water. Most people prefer to camp within easy walking distance of water, for the obvious reason that most of us need water to cook and drink. If you are worried about flooded campsites, there is plenty of room in the canyon for the river and dry spots along the sides for camping.
What season is best? This canyon is justly famous for roaring waterfalls. Those are at their best in the early summer, when there is plenty of snow up high to feed these waterfalls. But early summer on this trail can mean some wet feet, and even some tricky stream crossings in places. Later in the year the waterfalls are far less impressive, but that also means a much drier trail and fewer mosquitoes.
Of course, the actual flow in the river and the waterfalls will depend upon the snowpack and the temperatures. A warm day in late spring or early summer with have both roaring. By mid-summer those conditions are less likely.
As for heat, it can get warm in Pate Valley any time between early May and late September, so your best bet is to watch the weather and avoid heat spikes if you see them in the forecast. If you have to plan far ahead (and with the reservation system currently in use, I can understand why you would want to do this) you are probably safest picking dates that work for you and adjusting to the weather you find. That may mean hiking early and late in the day, and spending the warmest hours sitting on a rock in the shade dangling your feet in the water. There are worse ways to hike.
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