Chinook Pass is a common local ride, I think mostly for the views and climbing. When I motorcycled over Going to the Sun Road last year it reminded me in some ways of Chinook Pass. From what I remember Chinook is nearly always open by the end of May. That said, the shoulders can be narrow and the road is curvy with some switchbacks toward the top.
There are lots of alternative ways over the mountains, including HWY 706 to the south through Mount Rainier National Park. I'd choose HWY 706 -> HWY 123 then north to HWY 410, unless Chinook Pass was specifically on my bucket list (as it is for some). On the east side of Chinook Pass you can camp almost anywhere and there are good shoulders.
Once you get down out of the mountains, Eastern WA can be remote desert. Be prepared for long, flat stretches that go on forever, heat, goatheads, etc.