Yes, minor children employed by their parents are exempt from SS and Medicare tax.
There is no tax due if wages are less than $12,000. But you still have to withhold tax according to the IRS withholding tables.
Yes, you can issue a w-2 under your SSN. However, W-2s had to be filed with the Social Security Administration by January 31, 2019. And you must have paid the wages in 2018 to deduct them.
Please see this IRS document for more information:
https://www.irs.gov/businesses/small-businesses-self-employed/family-help
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