QIf I select an engaged enemy follower with this follower's UB On Evolve ability, will it refresh at the start of its controller's next turn?

ANo. Although the UB On Evolve ability's "engage it" effect will not apply to a follower that is already engaged, the "it doesn't refresh during its controller's next start phase" effect will still apply.

QIf I can't select an enemy follower on the field, does this follower's UB ability still count as executed?

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