grandparent watching kids - how to file 1099 and taxes

I have 2 questions. I hope to have my mom watch our kid (her grandchild) at her house in NJ and to pay her for it. We live in NY.  My understanding is she would NOT be a household employee since she would be watching the grandchild at her house. I also understand I have to issue her a 1099-MISC. 

Questions:

1) Mom is in NJ (where she watched child) vs. I live in NY - do I send 1099-MISC to both IRS and state? if so which state (NJ or NY)?

2) How does mom file her taxes so as not to pay self employment tax - would this just go on schedule 1 line 8 Other income or does she need to still file self employment? (she is not in the business/trade of daycare - as per: https://www.irs.gov/businesses/small-businesses-self-employed/family-caregivers-and-self-employment-...)

3) Is it still the case that mom will NOT pay FICA tax either (i.e. like in the parent exclusion of nanny taxes) if it's  schedule 1 line 8 other income? So the only tax she would pay is state/federal income tax?