It depends on whether you are receiving a tax benefit for the refund portion of the state income tax deduction. If that refund amount paid in 2019 and refunded in 2020 is reducing your federal taxes in 2019 it is taxable. Otherwise, it is not.
Yes, claiming the standard deduction will avoid any state refund from being taxable. Another approach would be to opt to deduct the general sales tax as an itemized deduction, instead of the state income taxes.
See this IRS article for more information about how the taxable portion of a state refund is calculated when you itemize deductions.