давай друзьями is incorrect Давай будем друзьями is correct.

"Давай останемся друзьями?" - it's better, but it sounds like a question. "Нам лучше остаться друзьями" - I would say this form.

"давай останемся друзьями" -is more correct

wonderful explanations, Irina. Thanks!

"давай останемся друзьями" means "let's stay friends" assuming that, "давай останемся друзьями" can be told: 1) to your "ex-love" - when you want to keep with him/her good relationship, but don't want to date with him/her anymore; 2) to someone who insists on becoming something bigger than just friends - when you don't want that kind of relationship with him/her; 3) to your real friend with whom you had (or are about to have) a fight - when you want to forget about that fight (or to stop that fight before it's too late) and keep the friendship. "let's be friends" - when you are offering friendship to someone - sounds in Russian "давай дружить" (this one is better) or "давай будем друзьями" (a little less common but okay as well)