I really hope they don't focus on people mistrusting and/or hating Superman in the sequel. That's the "Marvel" way of storytelling and even at that, they avoid using it in the Avengers
-verse movies. Iron Man is a celebrity and after saving New York, it doesn't seem as if the Avengers
are viewed as anything less than heroes.
I understand introducing Lex and having him try to turn people against Superman, but I think there are more interesting ways to have him be a credible threat. What makes Lex work in the comics is that, even though he personally hates Superman and what he represents, but he doesn't try to whip that resentment up in the public. That helps him maintain a positive image amongst the public and it serves to create more of an impact when he manipulates things to show Superman aiding him.