Recording calls and video conferences can be of great help if you, for instance, want to share what was said and done afterward. Now it can be done through Skype, which means you don't need any external microphones or other equipment.
I don't know; I haven't tried this feature myself yet. My guess is no, since that would not be shown when you record using iOS built-in screen recording feature.
I think the other party should have the right to know if you're recording. It's things like that why inappropriate and private images are often released.