Stormtroopers only miss when they’re ordered to miss. Han was an idiot at that point of the movie, should have guessed like Leia that they got away a bit too easily, and if the Falcon is so fast he should have led the Death Star through hundreds of jumps before he popped back to the rebel base. He should have been doing things like the Han of ESB that made star destroyer generals look like privates.
Engine light on isn’t always scary, just means something has disturbed the normal course of things and might need some looking into. A flashing engine light however, that can be worse that even the low oil pressure (which is bad).
Battery just means your alternator is screwed up somehow. I’ve driven mine like a EV before with a charger along for the ride to charge back up. Not great on the battery long term, but for short trips it’s doable.
Temp - well, that could be anything depending on if it’s an idiot light or a gauge. See the needle on a gauge go up quickly, same rule as the flashing engine light. Pull over and shut down.