Engines: the engine's power decreases as it takes damage. So an engine at 50% health would operate at 50% effectiveness, for example.
Guns: as they take damage, their rate of fire, turning speed, and reload time all become slower.
Balloons: the more damage the balloon takes, the slower the ship can rise and fall.
Armour: technically there's no difference between a ship at 1% armour and 100% armour; the one with 1% is just closer to having its hull exposed.
Hull: the ship doesn't behave differently when it takes hull damage (except when it dies, obviously), but some visual aspects change (bits get torn and bent, no effect on how the ship runs).
Waiting around next to a component on cooldown won't make it go faster or anything like that. Once you put it on cooldown, you should definitely run to nearby components that need repairs. There are some notable exceptions. The biggest one is the armour. If you repair the hull armour with a mallet stroke and you think that it'll still break before the cooldown is over, you probably want to stay there so you can get on the rebuild as soon as possible. It's all about judging the situation you're in.