i believe the road to sunrise at mt. rainier is the highest paved road in the state at ~6,400 feet.
AFAIK, the highest "pass" (i.e. road up and down both sides) is Hart's Pass at around 6,100', but gravel road. there are plenty of FS roads that go higher but most are dead ends.
in terms of riding hwy 20 unsupported, marblemount - washington pass - marblemount is a reasonable day (regular people reasonable, not randoneur reasonable) or you can ride over to mazama or winthrop one day, stay in a hotel and ride back the next day. In addition to the places listed, during the summer, you can get water from a drinking fountain at Newhalem, Diablo, Colonial Creek campground and the Washington Pass overlook. But in addition, remember that in the Cascades, there's always water around. The highway crosses hundreds of creeks, you just need to take some precautions to make sure it's safe to drink. i usually carry a tiny bottle of water purification stuff (iodine tabs or aqua mira drops). I almost never actually use it, but it's good to have just in case.