Yes to both questions.
This assumes the child lives with you more than he lives with his other parent. That is, only the custodial parent may claim the Dependent care Credit (DCC).
With only one child your may only claim $3000 of expenses, for the DCC. But your GF must declare the whole $7000 as income.
You can treat your GF as a Household employee and do all the necessary paper work. See https://ttlc.intuit.com/questions/1901554-do-i-have-to-pay-taxes-on-what-i-pay-to-my-household-emplo...
OR
She can report the income as self employment, if you two can agree on that.
