The most common reason your ships will appear as if they are not engaging is that they have no targets in range. The battle viewer will often show ships as advancing in combat when they really are sitting way in back out of range of every target. Watch the ship's speed... If the speed is listed as zero, it is *really* sitting at the back of the battle, not advancing as the battle viewer displays.

During a battlle, click on one of your ships that you think should be firing but isn't, and look at what ship it has targeted and what ship it has listed as defending. Also verify the ship is actually a capital ship and not an escort or guardian class. Those ships will not advance if they are guardiaing a ship that has not yet started moving forward.

I think most of us, certainly myself, need to spend more time evaluating the various ship roles so that we can utilize them more effectively in combat.

There are also ship components that can be attached to ships that give them various advantages and allow them to behave differently.  I hope someone will do a guide regarding these two subtle aspects of ship-to-ship combat.

For those of you who explicitly want to know what each type of ship does during combat, here is the link that explains it...

http://galciv3.gamepedia.com/Ship_roles

Everyone should read this. 

I havent played carriers.  I have fought again many however, the AI truly likes them once they get to that tech.  So, they are SEVERELY flawed IMO, and here is why.  Carrier modules have lots of value and very good threat.  However, the threat is MEANINGLESS, because the real threat are in the fighters which have already launched.  Remember, there is no retreat in combat.  The VALUE is MEANINGLESS as well, because if you kill the fighters, it wont have fighters next turn (i believe I have read you have to respawn them at a starbase?) and it will be easy to pursue the fleet and kill it with another fleet.

So, it pisses me off to high heaven, that when I am up against a lot of fighters, I wax them EASILY.  However, once the Carriers and Capital Ships come into range, all ships direct fire on them because of their high VALUE.  KILL THE FARGIN FIGHTERS!  I have lost so many ships because of this, or get damaged and I have to retreat to repair.  Now I havent looked that the details, but the AI Carriers have 'some' armor and weapons.  I'm not sure if the devs designed their carriers this way, to be of a higher threat than and individual fighter, so again, to soak up damage while the fighters continue to nail you.  

For having zero control during the game, the devs have a pretty darn good system I think.  However, it is flawed as well in the fact of focus firing on weak targets that can be destroyed in 1 salvo, like fighters.  If there are 10 fighters that have X threat and Y HP, and the carrier has 2X (from guns only, not the threat added from the carrier module) threat and 5Y HP, it makes sense to kill the fighters once at a time, because you reduce the overall threat quicker. 

It would be a different story if you damaged weapons, etc, while you destroyed HP, but that is not the case.  A ship is functional even at 1 HP. So, the proper strategy would be to eliminate the THREAT points as quickly as possible.   Hence, kill the fighters.  

I simply feel that in combat, the THREAT of carrier modules is meaningless, and VALUE is only valid, if its gonna be a close fight.  Meaning, if you have the option to kill half the fighters, or kill the carrier, and you are going to lose anyways, its better to kill the carrier.  But when you have superiority, killing the high VALUE target first is meaningless, because you kill them all in the end regardless, its just a matter of how much damage you take in the process...and you should kill fighters first.  ALWAYS.

There are modules which provide evasion. These start with the jamming modules. This is probably how you are gaining dodge.

Yes Thrusters do add to your 'agility' which can also increase miss chance. 

I think if you change your 'capital' ships role to 'guardian' they will target the interceptors and assault ships first...(High threat, low value targets) Hovering over the role in game will give you a brief idea of what ship roles each role will tend to target first.

However, it might be a good idea to only do this with one or two ships in a fleet. So essentially, copy your capital ship's loadout to a new design, and assign it guardian. Mix a couple guardian ships in with your fleet and they SHOULD tend to target the ships targeting your capital ships.