It may have been possible for most states years ago, but no longer due to fraud/security reasons. This article goes into more detail:
You will probably have to paper-file but I'm not sure how that works in states that mandate e-filing, like New York. If you're filing NY state, wouldn't hurt to walk through the e-filing in TurboTax, it'll let you know what your options are.
No. A state return cannot b e-filed unless the federal was e-filed. You will need to mail the state. Federal and state returns go to DIFFERENT addresses. Read the mailing instruction page for where to send your state.
When you mail a tax return, you need to attach any documents showing tax withheld, such as your W-2’s or any 1099’s. Use a mailing service that will track it, such as UPS or certified mail so you will know the IRS/state received the return.