No, Social Security benefits are not taxable on New York State individual returns.
Please see this NYS website.
P.S. I was answering your second question. As for your first question, some pension in NYS are taxable. Therefore, you have to indicate the source of the pension to show that it is nontaxable:
When you enter that you are a New York state resident, after you enter your 1099-R for the NYS & Local Employees Retirement, there will be a series of screens. One screen will ask you to Select Your State.
When you select New York, the next screen will ask "Where is this distribution from?"
This will give you the opportunity to select one of the options that is not taxable. Check this, fill in the boxes at the bottom as necessary, and this should remove this pension from NYS.
[Edited 2/19/2020 11:44 am CST]
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