The Walking Dead handled this topic. Shitty people will still be around, but their opinions don't matter anymore. The only thing that matters is working together to survive. And as much as you say that it takes an institution to enforce equality, it takes an institution to hold a group of people down as well. So unless the survivors are mostly racist/sexist, it really won't be an issue. It's not human nature to discriminate, that is a learned trait.