You do not need proof to file the return, just need it should IRS ask for it at a later date.
As long as your mother does not claim you, you can claim yourself and your child.
TurboTax will also ask about Head of Household, however you have to had paid over 1/2 of the costs of maintaining the household where you and your child are living. So you might want to determine how much you paid for the home versus your mother to claim this.