Do defense drones shoot down missiles before or after a dodge check

The Zoltan Shield will absorb bomb damage (as it blocks teleporters). This is only available on the Zoltan Cruiser ship, though.

I've found that the enemy AI tends to put their bomb weapons low on the list when prioritizing weapons, so damaging the enemy weapon system will likely take them offline. As they have low power consumption, though, it's easy for them to repair and repower.

Other than that, bombs are unique in their ability to bypass shields and drones. As you've noted, evasion helps, but otherwise there's nothing you can do but repair the damage.

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.