You can go camping in the Phou Khao Khouay national park north of Vientiane. In the Nah Ha park, you can stay in some bamboo huts. There is also a campsite at the Tad Leuk Waterfalls. I am sure there are more.
So to answer that part of your question, there are campsites.
Sleeping outside in an unprotected area by yourself can be tricky by the way since there are some poisonous snakes in Laos. I would be careful. This page here also states that it is dangerous but they think you might get robbed, specially inside the national park mentioned above. This site here says it's tolerated in some areas, but not actually legal. I read about bikers riding through the country and camping where they want to. Since a lot of places do not have any police coverage, and since they try to promote tourism, I think outside of cities you should be fine, but then again, there is the security aspect.