Non-Attack Damage

What counts as Attack Damage and what doesn't?

Attack Damage is damage dealt by the attacking character as a result of an attack. Any other damage, including damage dealt during an attack because of an ability, is non-attack damage. 

Say for instance that you were playing a full team of Aerialbots, with four planes in vehicle mode. Your opponent played a Strafing Run last turn and spread one damage to each of your characters. Your opponent has a Defensive Driving attached to one of their characters. You play a Bombing Run to move one damage from each of your planes to that character. What happens? Is all damage from Bombing Run dealt with a single damage at a time, per plane you have, or is the damage tallied and then moved in a single chunk? In other words, does that character take three damage from Bombing Run thanks to Defensive Driving, or none at all? Does Defensive Driving block Bombing Run at all, or is "moving" the same as "taking damage"?

Moving damage counters does not count as repairing or taking damage. So Defensive Driving would not block damage in this scenario. Maybe take a look at Bumpers!

Related to that point, does Motormaster block Bombing Run?

No.

Does Safeguard work if something tries to move a damage counter onto the character while it’s undamaged?

No. Moving damage counters doesn’t count as damaging the character, although after the damage counter is moved, the character will be damaged and Safeguard will no longer work. Sneaky!

Does each damage counter moved by Dinobot Sludge's Alt Mode ability create a new instance of damage, or does the total amount you want to move count as one instance?

Neither. Moving damage counters does not count as repairing or taking damage.

Does Peace Through Tyranny do damage to a character when it is K.O.'d or does it just move to the K.O. Area without doing damage? 

It K.O.’s the character without damage, but characters in the K.O. Area are assumed to have their maximum damage.