OP costs change with hull sizes. So a cap would have a higher cost while smaller ships would have lower costs.
Not for Operation Center, it's 30 OPs regardless of hull size. I found it best installed on a "passive" ship like a carrier, or better yet, strap it on a shuttle and let it sleep on the edge of a map (or capture unguarded strategic points, since the AI struggle to do that). If you're planning to keep going with that "Admiral" run idea, Operation Center is invaluable.
I very recently did that, a whole campaign with autopilot only. And while early-game was extremely painful, the strat start to shine during mid-game where you can field a proper fleet. It will unfortunately start to show its limit against end-game opponents (late IBB, etc.), as your deployments point capacity will shrink depending on the enemy's size. I'm still working on some potential solutions though...
A few tips : you can't go with a defensive fleet playing that way. As John said, an agressive player can offset a passive AI fleet, but without your intervention, the battle will devolve into a CR war, which you really want to avoid reaching a certain point. It's why favouring agressive loadouts tends to be important to take advantage of the enemy's weakness. Combo this with elimination orders to quickly take down targets that stray too far from the main pack.
Another thing, while your destroyers and cruisers are the bricks forming the main part of your fleet, fighters and bombers are the mortar holding everything together. Fighters are extremely flexible and can bring support nearly everywhere quickly, allowing you to wither down an opponent, finish it, or even save an ally. You won't get really far without a bunch of carriers.
Also, you can use control groups (ctrl+#) to create custom squads in your fleet. Very useful when you have to micromanage more than 20 ships.