Movements, shields and weapons
I agree that the carrier should have a consequent size and low mobility, but I'm not sure it sould have a Dreadnought level of shield protection. The carrier when caught off guard (and without any allies to defend them) should be easy target for the other ships, and so have less armor and shields than most of the other ships.
For weapons, I don't have much to said, maybe a little bit more firepower than the artillery secondaries (but not that much).
Fighters and devices
A.I. controlled fighters/devices seems the easiest way to implement them. However maybe you can add more interactivity than "point and forget", with orders and formations that can be given in flight.
Your second idea with map interaction is really too much, it fits more for a commander chosen by the players rather than a kind of ship class/abilities. It's not a carrier at this point it's the Capital ship ! Not to forget that you can have multiple Carrier ship, so how a same map device (like a turret for exemple) will issue orders ? If two players try to get control of it, it will be really frustrating. And if you have a really annoying person at the Carrier command I can't imagine the nightmare of the pings and orders spamming.
For the last one, it could be interesting, but if you launch swarms of fighters how will you determine which one you'll pilot ? How can you continue to use the carriers abilities and ordes and use the fighters abilities without constantly switching beetwen the two ?
Actually I see the fighters/bombers/devices placed as ability slots. So you can pick 4 types of "hangars". Sending squads or formations and loosing them will immediatly activate a full cooldown (which should be rather high, so you have to use them smartly) until you get a new crew. However I'm against the "you totally loose this ship until the battle end" thing.
I'm not entirely sure I understand your last point. Do you mean that each time you issue and order to a squad they just warp in and then warp out ? If yes, then it will be next to your Carrier, cause if they just warp directly at the opponent face, it will make them far too efficient.