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.