If you know your work is worth more, talk to the family and negotiate a payment that meets both your needs.
If you are interested, just ask the family for clarification.
If another job presents itself, just explain that you need to finish your 2 week resignation commitment. If you need to give less than 2 weeks notice, I would suggest that you bring that up with the family as soon as you know you will need to leave in less... more
That depends on the sitter's policy, and if they need to provide direct care for you.
Professionally explain that you had an occasion that went badly.
You need to be firm and consistent, and you also need to follow through. Stick with your answer, and always give the same answer. Consistency and follow through are key, without those you won't have respect or control.
Yes, you are allowed to include volunteer hours.
I would suggest talking to her pediatrician.