Residents of Westchester County are not subject to the New York City income tax.
No, in general (New York City employees who are nonresidents are required to pay the city tax), only New York City residents have to pay the New York City income tax. You should contact your employer to have your city withholding stopped if you have not already done so. See Personal Income Tax and Non-Resident Employees.
It's a nitpicky point, but NYC claims that non-resident NYC employees don't actually have to pay the city income tax. Instead, they must - as a condition of employment - pay the city an amount that just happens to be the exact amount they'd pay in city income tax if they were a resident.
Non-resident NYC employees make that payment with Form NYC-1127 (which TurboTax does support):
https://www.nyc.gov/assets/finance/downloads/pdf/24pdf/business_tax_forms/nyc-1127_2024.pdf