Tactical combat decisions

Not sure if you already checked, but under empire options there is the ability to set what the AI attacks first/at what range. I have my fighters set to attack other fighters first, then satellites etc, while my bombers go for cap ships.

Of course it's not perfect, for some reason my fighters still attack things other than fighters first some of the time. But it's a start if you haven't already messed with it.

Ya, edit your strategies.
i don't know what the computer does for strategies. but first thing i do in all my games is set up my strategies.
i also like watching the combats. but i want my ships to fight the way i want them to without having to control them... at all.

Use FIGHTER ATTACK for fighters... and set them to attack Smallest and Fastest at the top of the list. they will attack Fighters, Drones and Satellites first. Leaving your cap.ships to duke it out with the Enemy cap. ships.
The other thing you can do is set Target Types. then move everything around into the order you want your fighters to destroy things. ei. Fighters first... Fleeing cargo ships second... mwa ha ha ha ha.
If you set all your strats right, your fleets will do exactly what you want them to without you having to intervene.

Fighter attack is meant for fighters, or for carriers (attack via fighter)?

Carriers, I believe. Fighters are just supposed to use Optimal Firing Range or whatever, like a normal combat ship, but I have mine with their own custom strategy that makes them go after fighters first.

Interestingly enough my fighters still won't go after fighters first, even after I set target priority up as Target Type (fighters at top of list), then Smallest then Fastest. They still insist on going after ships/planets before other fighters. I had to split my fighters into interceptors and bombers and then remove ships and planets entirely from the interceptors target list (so they'll never go after either, but they WILL hunt down drones first then fighters then satellites).