Double negatives in English are a complex topic. The simple answer is that double negatives are incorrect in Standard English forms. But, they can be a correct way of forming a positive in Standard English, or a correct way of forming a negative in some dialects of English. You should avoid using them, but you might see them used in media or in conversations with English speakers.

They are both correct, sorry if that does not help. "Do nothing" could be used as a command but "didn't do anything" is more frequently used. For example you could say..."what has the boy done?" ....."he didn't do anything!" As for "didn't do nothing", I wouldn't say it means you're uneducated but it is very common in England and is basically wrong, but it is still used a lot! I wouldn't recommend you use it in your studies because it is not correct, at least in this context.

"Didn't do anything" is correct. "Didn't do nothing" is slang and is considered "uneducated".