You have several questions. Please note the following:
Payroll taxes. There is a nice provision that allows for a child working on behalf of a "family business" to be exempt from payroll taxes. However, as you will note at this IRS link Children and Payroll Taxes (and notice the section in purple), this treatment is not available if the family business has an S-Corp or a Partnership structure, unless (only in the case of the Partnership), the only partners are both parents. Otherwise, FICA, and FUTA withholdings are required.
Unemployment taxes. Your state may require these to be paid regardless. Check with your state guidelines on this.
Deferred Social Security taxes. That warning does not apply to your situation. It is absolutely possible that you may have to pay the standard Social Security/Medicare taxes according to the first paragraph, but deferred Social Security was a COVID provision brought into play through executive order in 2021. That has nothing to do with your situation. If that diagnostic is preventing you from filing payroll taxes or other returns, please call in this experience to our phone customer support service by using the instructions in the following link: How do I contact TurboTax?
New York is Charging Tax. Why? You may not be required to withhold New York taxes on your child, but that doesn't mean he doesn't pay taxes. Note this New York website: Filing requirements in New York. You will notice that an individual claimed as a dependent who has more than $3100 in reportable income must file. This usually means they will pay tax on amounts above $3100 unless they qualify for certain additional deductions or credits. Given New York's base tax bracket, $125 actually sounds like the correct amount on your child's reportable wages.
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