How to deal with hacking drones

The Anti-Ship Drone is armed with a laser, which does damage to a single room targeted randomly by the drone. It also takes the shields down one level if it hits a shield..

The Anti-Ship Beam Drone is armed with a beam weapon, which does damage to multiple rooms along the path of the beam, but which does not affect shields at all. Instead, shields affect the beam, in that each level of shield reduces the damage of the beam. Since the Anti-Ship Beam Drone only does 1 damage per room, even a single active shield can render the Anti-Ship Beam Drone completely useless.

As always, the best defense is a good offense. If you can take his drone control offline, you just need to work harder to keep it offline. That might mean targeting the drone system to disrupt repairs even when there are other undamaged systems to target. You are correct that drones can come back online quickly and this is an issue. I'm not sure the enemy actually has to spend drone parts to do this, though you would have to.

Defensively, dodge and shielding of course matter. You can't have too much of either (well, dodge from engines experiences diminishing returns at higher levels of system improvement, so don't overdo it there). Getting to 2 points in shields fairly early can be a very good investment, and I like to have 3 by sector 5 or so.

You can destroy drones around your ship with ship weapons, however this is a coincidental thing and the drones can be brought back online with the expenditure of a drone part. If the orbiting drone happens to collide with a projectile, boom. It's very unusual; I have only seen it once myself.