How do drones choose their targets

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.

Beams that do no "damage" are blocked completely by any shield.

kotekzot notes that a Zoltan super-shield (the green shield augmentation on certain classes of ship) will take 2 damage towards its 5 damage limit from beam weapons, however.

The easiest way to test this is with the Slug Cruiser - it starts with the anti-bio beam. Just jump until you fight a ship with a shield, and try firing the beam without any other weapons. Visually, the beam is blocked by the shield, and even after firing it 10 times at an enemy ship I wasn't able to kill even one of their crew.