1.To answer your first question, yes you can file as head of household with no income, and when you're the custodial parent.
2.However, the noncustodial parent may not claim the child for the purpose of claiming head of household filing status, the earned income credit, the credit for child and dependent care expenses, the exclusion for dependent care benefits, or the health coverage tax credit.
https://turbotax.intuit.com/tax-tips/family/guide-to-filing-taxes-as-head-of-household/L4Nx6DYu9
