What you're offering is very fair considering what you're offering. Although I think that 60 hours a week is a little much for anyone but if that person doesn't have anything else going on then I think it's very fair!
Parents support babysitter/nanny when disciplining. Many parents tend to give in or side with their child if they come crying to them that the babysitter put them in time-out rather than support the babysitter's position and reasons. It tends to show the child they can get away with... more
If she is going out of her way and using her own vehicle, then yes you should definitely pay for gas/mileage.
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