Q: Is this a tiger?
A1: No, it isn't (a tiger).
A2: No, it is not (a tiger).
A1 is an example of spoken English, whereas A2 is an example of written English. Speaker will use "No, it is not (a tiger)" so as to emphasize the negation. If I were teaching that structure, I'd use, "No, it's not." The reason being, contracting "is" and "not" is a jump ahead, grammatically.
Q: Happy New Year
A: Same to you. (If said with a smile, it'll be taken as an informal response)
If you're teaching that greeting, it'd be best to use, "Happy New Year to you, too" or "To you, too" or "You, too." It's a good idea to start with the Standard.