Normally, you may be able to avoid tax in Idaho if you work in another state, but since the state is Washington which has no personal income tax, it will not work for you. You would get credit for taxes paid to another state on your Idaho return, but since you paid no tax to Washington, you will have to pay your full Idaho tax.
A person who works in Idaho has Idaho source income and has to pay Idaho tax on it.
Unfortunately, TurboTax is not licensed to give advice on payroll tax laws, so I can't answer the SUTA question.
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