There is a KOA style campground on the west side of town, can't miss it.
Otherwise, you will have to "stealth camp", meaning no tent, no fire, etc, just sleep in your truck. I would suggest the following places to illegally stealth camp:
Lower Fish put-in
National Forest trailhead just before you get to Strawberry Park Hotsprings
anywhere a couple blocks off of downtown, not on Yampa St. or Oak St. though.
Chan's house (ha ha), but seriously, Chan probably wouldn't mind if you know him, he was talking about hosting a party Sunday night.
Danimal's backyard, it is big and on Soda Creek, talk to Danimal Pianimal
not Wal-Mart (its the only one in America that doesn't allow RVs/camping)
maybe up Buff Pass, but probably not, they're saying June 7
Yampa River State Park in Hayden (25 miles away)
mabye stealth camp over at Fetcher Pond, sorta in behind Staples, on the river, as you come in to town.
just off the head of Sleeping Giant, if you know where that is
random turnouts as you come down Rabbit Ears Pass, out of sight of the highway
You wouldn't be able to car camp, but you could hike some gear in past Fish Creek Falls and squat for the night.
All but the KOA and State Park are illegal.
The water is cold. Don't be fooled by the warm temps. I tried a shorty yesterday at the hole = mistake.
Also, it is supposed to dip down into the 60s for highs this weekend, lows above freezing.
As for class III, just put in on the Yampa at Mt. Werner Road for a long town canyon run. Take out at the "multi modal center" about a mile past the hole. The Elk, while having no real holes, or moves for that matter, would certainly punish a swimmer with a long, long, cold swim. It is fast and pushy. It is actually fun when it is above 5000, but you should have a good roll, whereas the Yampa is easy to swim out of, just ask Routter.
Unlike BV, Salida, CB, etc. we have a lot of cops who will be looking for any reason to pull you over after dark. Do not drink and drive, bring a bike. If you get caught, PM me . . .