You have several questions, so I'll try to answer each.
Do you need to file NY state? No, you do not. Since your wife is filing MFS in New York, and you neither lived nor worked in NY, you are not required to file there. Only she is.
Do you need to mail in your state returns? Probably. The answer is a definite yes if your wife is the secondary taxpayer on the Federal return. Massachusetts for sure requires this, because they require a mock return for each spouse filing separately to separate the Mass income to each. (And Massachusetts requires you to file separate returns because you have different residency periods in Massachusetts). And if you are the primary on the Federal return, the mock return your prepare for her separate filings cannot be e-filed, so the New York return will have to be mailed also. (New York "requires" e-filing but there are circumstances that cannot e-file that they accept by mail. This is one of those circumstances).
Do we need to include a copy of the Federal return in the state copies? Yes for Massachusetts, and probably for New York. Massachusetts requires that mock "married filing separate" copies of the Federal Return be included, so you will use the mock return for your wife to send a copy of the return to Massachusetts (this prints automatically). I don't know that New York requires the Federal return to be filed, but I do think it is wise to do so anyways.
Typically, TurboTax will provide you with the necessary documents for your state filings. If a copy of the Federal is included, trust that you will need it for your state return that you are mailing, and if not prepared, you shouldn't need it but should always keeep proof in case the state asks for it.
**Mark the post that answers your question by clicking on "Mark as Best Answer"