I vote for the Willamette
Gooz,
One trip that I'd like to do some time is the mainstem Umpqua River. It doesn't quite meet two of your criteria: It isn't really remote, and there are some class III rapids. Check out Tim Palmer's
Field Guide to Oregon Rivers for more details. Tim says, "Though virtually unknown as such, the main stem is not just a great Oregon canoe trip, but a great American canoe trip." Tim describes that while there is a road along the river, large back bends give the ~feel~ of being remote. So maybe it really DOES meet that criteria of yours. Tim says it's "too easy" for most kayakers, "too flat and windy" for most rafters, and great for experienced canoeists. He also warns that there can be blooms of cyanobacteria. This is obviously a longer stretch than you're looking for, so you could pore over some maps and pick out a two- to three- day stretch that would meet more of your criteria.
Another interesting trip that doesn't meet all your criteria is the Hanford Reach of the Columbia River. It's 51 miles, but camping is not allowed. You would have to do multiple shuttles and camp elsewhere. This is the last free-flowing reach of the Columbia. This one is also on my list for some point in the future.
Finally, the Willamette is often overlooked as a great resource. This trip actually meets all four of your criteria. Many parts of the river can feel quite remote, but you are actually traveling through the most populous part of the state. There is a
water trail, and on the river you can find mile markers and some camp markers. There is PLENTY of camping in the summer. Lots of islands. Visit the website to use an interactive map to explore the area. You can also buy waterproof maps. You should note that since the map was published, Norwood Island is no longer private property and makes a great camp. There's a shuttle service you can hire if you like. You can pick a trip of any length. If you are comfortable in class II (more like II-), put in at Armitage Park. There's a parking fee, but you can launch QUICKLY and then leave your vehicle across the road. The shuttle service will pick it up quickly and take it to a secure location and then deliver it the morning of your take-out. Depending how much you want to paddle, you can do the 50 miles to Corvallis in two or three days. The river flattens out below Harrisburg, but gets just a little steeper just upstream of Corvallis. Another ten miles takes you to Albany, also very realistic for three days. I'd recommend going all the way to Independence. That could take four days. I'd recommend NOT going all the way to Salem. The last section below Eloa are right along Highway 22. It's loud, and not so scenic. I vote for the Willamette for your trip. In fact, I'm heading out for a two-day from Coburg to Corvallis in the morning.
You WILL see other boats. Many people do get out to enjoy the river. But you'll mostly have the place to yourself, especially on week days. Plan on lots of tubers in the couple miles upstream of Corvallis. That's the only crowded stretch.
You might also check out the Rogue below the Wild & Scenic section. Tim Palmer says it's a beautiful canyon, but you will see LOTS of jet boats. Huge gravel bars. 32 miles.
Good luck!