E.T.
The federal government and most major companies reimburse at 50 cents per mile; that includes wear and tear on the car. 30 cents a mile seems pretty fair to me.
When we had a nanny, she drove one of our cars so we didn't have to reimburse for gas, but yes - I would have reimbursed if she used her own car.
Nanny rates vary depending on city and region, but from what I've read on here, some where between $10 - 15 hour is pretty typical. In addition to the duties you've already listed, you should expect her to be coming up with ideas of things to play with your younger son... and certainly no TV or texting while she's at work.