How popular is Superman?

There are moments where he has both fans and angry mobs in some scenes, and then other moments where he can stand in the street with Lois Lane and get a kiss without anyone bothering him. Can we go with one or the other throughout the film?
Why is Aquaman on a surveillance tape?

Its one thing to need some privacy, but going and attacking a camera straight on might attract some unwanted attention to his kingdom.
Why is Zod’s ship still intact?

Is there a reason why Superman would want to keep this spaceship intact in the event that it could be used against him later down the road? Based on his knowledge of how Kryptonian technology works, seems like he wouldn’t want to leave it lying around.