Normally we don't use the subject pronoun (you) in an imperative statement, but it can be used for emphasis. Be careful though if you use it on anyone who is not much younger than you and in your charge (eg parent to child) - it can sound very direct! For this reason, I don't know if Duo will accept it. :-)
That is correct. This is because the negative always follows the template of a formal command rather than an informal command, and so is subjunctive. It kind of makes sense because you are making a request of someone to "not do something" rather than telling them "to do something", but maybe that is just how I remember this rule.
I would say, in this context, that yes, they are equivalent. It's possible that "Don't do that" would have a slight nuance towards "Don't do that; do this instead" whereas "Don't do it" is pretty final. However, if Duo asks for "Don't do it", I've learnt to give it EXACTLY what it wants! :)
I think it is just that Duo has not put the contraction into the computer. It is correct.
Although, there is an unwritten(?) convention in English that a contraction makes the sentence a little more casual and to hear your mother or a teacher say, 'Do not do it!' you might be more likely to think, Oh boy, I'm in trouble. :)
Whereas the familiar voice is used with children, friends, and others of the same generation, the formal is generally used with those who are elders, teachers, professors, supervisors, bosses, and strangers – basically those with whom you are not on a first name basis, or even if they are younger and NOT in a position of authority or respect, those with whom you might want maintain some distance.
I doubt any disrespect would be inferred by any such persons if you were to say "¡No lo haga! / 'Don't do it!"if they were about to step off into a pit of vipers, a crocodile lair, or an open elevator shaft.