Josh: That sounds like a great trip, I can answer some of your questions but not all. There is a very nice campground in Riverside that offers camping and cabins at very reasonable rates. Riverside is a small town (wide spot in the road) that has a gas station, bar, restaurant and I think a small grocery store. Lazy Acres Campground and Motel, Riverside, Wyoming
Encampment is a little larger but not by much.
I would call John at the Four Seasons Anglers Fly Shop in Laramie. They do guided trips on the North Platte and ask him what he thinks about your timing, pre-runoff, etc. Wyoming Fly Fishing Guides, North Platte Fly Fishing Vacations, Gray Reef Trout Guides
You might also call the Stoney Creek Fly Shop in Saratoga and ask them.
Stoney Creek Outfitters - Home
I've fished out of Bennet Peak Campground and I know that if you drift the river it runs into private land, but not sure how far down that actually is.....John from Four Seasons can give you that info.
From my general knowledge of that area, if you run into private property it is very well marked.
I haven't done any backpacking into the Encampment Wilderness area so I can't advise on which is the best approach.
Late April/early May will avoid the runoff, I'm just a little concerned about the wx that time of year. Ask John what he thinks and get a second opinion from the folks at Stoney Creek Outfitters.
I can't lay my hands on my Medicine Bow NF map, but I think you are talking about the area near the headwaters of the North Platte river up in the North Park area. I have never been in there, but have heard that it has some incredible fishing in the fall (haven't heard about the spring), hiking in from the north end, there is a public access point on the north end of the canyon off hwy 230. I think the reason folks fish into the canyon in the fall is the low water levels, at higher water levels you can't traverse the canyon. That brings up another point, all the public access points are pretty well marked with brown signs along the hwy 130/230.
Also there is a really good fly shop in Walden, CO (the headwaters of the North Platte), you might also give them a call and see what they have to say as they also guide on the North Platte. North Park Anglers