If my daughter does not claim the Child and Dependant Care Credit, can I consider what she pays me as a "gift" and, therefore, not have to declare it as income on my taxes?
Yes. But I can't be certain how that may affect things on the legal front aside from taxes.
If my daugter does claim the Child and Dependant Care Credit,
a) how does my daughter classify me as caregiver when filing her taxes and what form(s) if any does she need to supply me with?
She doesn't need to provide "you" any forms. You are the one who needs to supply her with a statement of how much she paid you for the year, along with your legal name and Taxpayer Identification Number (your SSN). Your daughter is required to report who she paid, and how much she paid, in order to get the credit.
b) how do I classify myself (ie "self-employed" or "household employee") and declare the income when filing my and my husband's taxes on Turbotax?
It depends. You could be a household employee or self-employed. It just depends on the facts and circumstances. I suggest you start reading at https://turbotax.intuit.com/tax-tips/self-employment-taxes/tax-tips-for-nannies-and-at-home-care-giv...
Lastly, will the Deluxe Federal and State version of Turbotax be sufficient given our situation, or do you recommend another?
Deluxe should be fine.
