Should the Javeils be abandoned? They seem to die too quickly. Should I focus all of my ship caps on kodiaks and hoshikos? Are the command crusiers important?
If you're against AI, all frigate based dps should be phased out soon, other than flaks. Long range frigates have their use, and Illuminators might be good enough for late game fleets, but LRMs especially are quite poor. It's hard to even monitor their DPS progress due to the goofy trail of missiles and their travel time. The second command cruiser ability with +25% damage can be good if you micro it, but they're kind of tertiary priority.
I have a problem microing the battleships, because in a battle it gets very messy, and the capitalships are so slow in turning and moving, that it's going to take a long time for them to focus on to the target you want them to, I just use auto cast.
Yeah, the turn speed can be annoying, especially in the case of ones such as Dunov that have a forward firing main ability but turn on their side to attack. Many abilities are easy to micro, though, such as Flak Burst. Put the cap ships in the same hotkey group and use the tab key (and shift+tab to go backward) to cycle through the capital ships and use their abilities. This is a lot easier if you've played WC3 before

Turrets are good, but a problem is that they have a range, and if the enemy stays out of the range, they are powerless. Should I use hangars at all? They get shredded so quickly by the enemy bombers, and I already have planes launched by the 8 capital ships.
The nice thing about AI is they're too stupid to avoid them. Just put them on the side of the planet that they'll be warping in from. Hangers are rather useless for actual defense. I tend to use 1 per planet to chase off scouts/single frigs. You just don't get enough space for them in tac slots to keep up with the heavily carrier based ai fleets.
And are the light carriers useful? The AIs seem to use them a lot, but one squad per carrier seems a bit less.
They have their uses. Bomber fleets are great for razing orbital structures, and fighters are honestly better against strike craft than flaks. However, especially against some capital ships, strike craft can become completely negated within 1-2 minutes of a fight starting. You'll probably have an easier time not bothering more often than not.
Edit: Regarding the flak bursts, radiation bombs, and anything that has splash, don't your own ships get caught in them too?
No.