Ps4 – How to switch to the other side camera of the character

I can confirm that tag progress does carry over, at least from one game into the new game plus. I don't know whether the same holds true in an entirely new game.

No, you'll have to begin and finish the game on Hard to unlock those bonuses.

Even if you finish the game on Hard, starting it on Normal will reward you for finishing it on Normal. This prevents players from gaming the system by changing the difficulty in the last moments of the game.