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With 3 months available, I don't think you have to cross anything off your list - including Yellowstone. You can drive from coast to coast in about a week, so really everything is available within your timeframe.
Well, probably is a bit too harsh of a term. The pass typically opens up between mid-may and late june, so it is entirely possible that it would be open by the time you travel. The last few years, the pass has been opening in May, because of a lack of snowfall in the mountains. There's just no way to know what this winter will bring, but I certainly wouldn't write off the possibility.
That shouldn't be a problem at all. The earlier in the season, the more likely you are to see higher, and faster, water - because of snow melt. It will also depend upon where/what river you end up choosing, but there should be options all summer long.As far as I know river rafting (Colorado River for example) should be possible then, correct?
Really, if you're traveling from June to August, you're going to be in the peak of the summer season for pretty much your entire trip. It shouldn't really make too much difference, although I might opt to go east to west, to ensure that mountain areas of the west are completely open by the time you get out west.I know I will arrive in august on the east coast (probably high season too). But I assume SF to NY is better than NY to SF considering the months I'll travel.
Another option I would give some serious consideration to would be to do a complete loop. With 3 months, you've certainly got the time to do it, and it would let you easily see things in both the north and south parts of the US. It could also reduce some of your costs, since you'd eliminate the one-way drop fee on the rental car.