QIf my field is full, can I still play this follower's UB On Evolve ability? Will it count as executed?

AYes, and yes. Since your field is full, the PriConne follower you select would remain in your cemetery.

QIf I choose not to pay the cost of this follower's UB On Evolve ability or can't select a follower in my cemetery, does its UB ability still count as executed?

ANo. The ability wasn't successfully played, so it doesn't count as executed.

QIf a UB ability I control executes during an opponent's turn, does that trigger this follower's "whenever a UB ability of another follower on your field executes" ability?

AYes, it does.