Can ships docked inside other ships (like capital ships) still perform their functions without taking off i.e. drone control, scanning, and shooting out the hangar door?

So yes, depending on what exactly it does. I'm not going to say yes to everything. But for the examples there, controlling drones, there's nothing that's going to stop you there. Shooting out the door, nothing going to stop you there. In the instance of, say, a scanning ship that is docked inside a capital ship, that's probably not going to work as well as it would if it was out in open space because naturally it's inside a great big hulking lump of metal. So yes, within reason.

I imagine even if it does work during the first implementation, the early implementation, that does seem like something in a diegetic universe you'd want to impair somehow.

Yeah, you don't want to have, say, 50 Terrapins inside a Bengal hulk being pushed around as some sort of uber-scanning station.