In your case, 3a is true but 3b is false, therefore you don't fit into the exception requiring payment of HHE tax, and your mother is exempt.
However, you still need to issue a W-2 to your mother at the end of 2023. There are a couple of ways to do this online for free or a small fee.
And yes, you can claim the child care expense for the dependent care credit, assuming you meet all the conditions (must both work full time, children under age 13, and so on.)
