Both are correct but at least in the USA -- or at least the area I live -- just using "you too" sounds more natural. Also "(the) same to you" can carry a bit of a negative connotation as I've typically heard it, and used it, when not having a pleasant exchange with someone. It's a bit like throwing an insult back at the person who first said something. Oddly, perhaps, people will say "you do the same" and it's a nice positive response to that "Have an nice day".

This is a more formal way of saying it, but people may know you are not a native if you say it like this. In english, there are usually shortened responses to greetings. For example, in America, it is more common that someone would say "you too", as a response. Also, "same" would work (but if you say "same" deliver this with a smile, because any response with one word can seem dismissive or rude if said with a serious face). Here are some other more formal, but less common responses: "yourself as well" "you as well" "the same to/for you" "likewise"

Yes, that is common. We sometimes remove the word "the" and simply reply "same to you."

Yes absolutely. Or you could even just say “thank you, and you too”