Is this for the determination if it will qualify someone as your qualifying child, or what is the purpose of the question?

To determine if home heath care is a medical expense that can be deducted.

Do you have to pay employment taxes on a nonfamily caregiver if they are private duty and not thru an agency in this case?

Likely, but not a guarantee.  See this website: <a rel="nofollow" target="_blank" href="https://www.irs.gov/businesses/small-businesses-self-employed/family-caregivers-and-self-employment-... is a link on that website for Publication 526 (Household Employees) that goes into greater detail.  If you are required to report the taxes, these will be reported on Schedule H if you have not withheld them through the year.  But look through the materials, and feel free to ask additional questions.