고추 is technically a fruit but like tomato it's used as a vegetable. Many English names for different kinds including, chili, chilli, chili pepper, bell pepper, capsicum, etc.
No, 후추 is black pepper.
No idea if Korean has different words for capsicum/bell pepper and hot/chili pepper, but they're basically the same plant so it wouldn't surprise me if it didn't.
Bell pepper is 피망. And actually, considering how much food permeates Korean language and culture, I'd be surprised if they didn't. They are still very farm-oriented and, living in a country that has been devastated by war again and again, they'll eat just about anything. Even McDonalds.