I searched for the answer to these but very few results came up and it looked like the topic title was the only thing searched.
1. How much damage does a fleet need to take before it reaches the maximum bonus for speed, damage, and defense?
2. Does the fleet damage carry over? Will the fleet go into the next battle(s) with the same bonuses in effect?
2. The crews only check the current missing ship health. It does not matter when or where they lost it.
If you're asking about the maximum damage needed in regards to finding out what 'the best' crew is, Grease Monkeys are basically unmatched at around 30% health. Keep ship stacking in mind though, it might be a detriment to use these crews in certain targets.
In the last 5 minutes, send your dead fleet to dock
Set repair do NOT cancel it
GM will still offer 1/2 repair althoug it expires, but will still function as normal so long as the fleet is slow repairing