QWhen playing this follower's On Evolve ability, if I can't select an enemy follower on the field, but Overflow is active for me, will this follower still have “Once per turn, when this follower deals combat damage, refresh it?”

ANo. If you can't select an enemy follower, you can't play this follower's On Evolve ability.

QOverflow is active for me and I have another Kagero follower on my field, giving this follower Storm due to its passive ability. If I use this follower to attack an enemy leader on the same turn I played it, and an opponent destroys my other Kagero follower with a Quick card such as Execution, will my attack still go through?

AYes, it will.

QDo I resolve Twin Drive before an opponent can play cards and abilities with Quick?

AYes. When attacking, all abilities that are triggered must be resolved before non-active players can play cards and abilities with Quick.

QWhen attacking using a follower with Twin Drive, if I reveal a Trigger with the first drive check, do I resolve the Trigger before or after performing the second drive check?

AYou resolve the Trigger first. For example, if you reveal and resolve a Draw Trigger with the first drive check, you will draw a card, put the resolved Trigger into your cemetery, then perform the second drive check. If you choose not to resolve the Trigger, you will put it on the bottom of your deck, then perform the second drive check.