Will drones continue to operate / fight if the targeted ship capacitor is reduced to zero?
Yes. Drones are unaffected by capacitor issues.
Another useful aspect is they will keep attacking the target even beyond drone control range, up to like 500 km or something ridiculous like that, though once the target dies they will fly back to your ship and can only send them to attack if the target is within drone control range (regardless if you manually order them to or they auto-aggress said target) if I recall correctly.
Also obviously they never run out of ammo.