What's the lowest deal per hour for caregiving aid?

California's minimum wage is $10/hour so that is the absolute minimum. As an employer you have to also pay your share of the SS tax. If you are not the employer directly but hire through an agency then they take care of all the tax and overhead expenses. If you hire someone as a self-employed "contractor" expect that they will charge accordingly to cover these payroll expenses.

From what you describe you need a caregiver and a maid. I would expect at least $15 an hour around here in Birmingham, maybe more, to have both duties performed.

That would be what the Agency and you agree upon as per your employment contract. I would believe that an Agency cannot go below the State minimum wage for your pay. What the Agency charges the client is much higher because of the expenses of owning an Agency, being licensed, bonded, insured, and having workman's comp insurance. Then the overhead cost of having an office and office employees.