So, does the unit(s) you have left within an outpost being razed have to stay in the outpost for the whole 3 ticks in order for it to become a camp?
They do not have to.
As long as you have something "attached" to the outpost being razed/camp, they camp will not disappear. This includes attacking squads, defending squads, moving squads, spies, agents, and nukes.
As a follow-up, once the camp is already established, do you have to keep at least one unit stationed at that camp in order for it to remain?
You do not, an agent or a squad moving there/away will also keep it there.
Will the camp disappear if you move the unit(s) away from the camp?
If you remove everything "attached" to the camp, it will disappear.