Rand could not channel a large amount of saidin in his sleep, no more than anyone else can. We don't even know for sure that a channeller can embrace the source while sleeping, of if he/she must do so before going to sleep if she/he wants to channel Spirit while asleep. Touching a Stone (the steps actually, Rand was touching the steps) isn't enough to use the PS either, you need to channel into one of the symbols.

Clearly, that's not Rand who activated the Stone. Lanfear did. She's the one who isolated him (as much as she could, she had no choice but to bear with Hurin and Loial being transported too) to begin her campaign of seduction.

It's probably not Rand who brought them back either, he had no idea how before Verin told him. It's Lanfear again. She made Rand believe he did it, to hide the fact she could channel.