Damaged starships will automatically repair themselves when they have supplies on board. 5 kt supplies repair one damage point.
Rebel carriers will automatically build fighters as long as they have the appropriate amounts of minerals available (ShipFighterCost).
Fueled ships which reach a planet which you don't own will send you an exploration report.
Some ship functions will permanently operate, for example, anti cloak or alchemy.
If a ship is moving faster than Warp 1, and ends up close to a planet, the planet's gravity will pull it towards the planet. See AllowGravityWells and friends.
Ships may be configured to require fuel for staying on duty, see FuelUsagePerTurnFor100KT.