Go out. Playground, walking, library.
I do not know the cost of living in MO but $12/hr does not sound fair to me. And more if you will taking care of 1 toddler and an infant. I know what is like and nope. I would go at least $15/hr.
That would depend on age of the sitter, duties, experience, number of kids and also the location. Where I live I dont work for less than $15/hr.
People micro-managing my time.
It is supposed to be a bit higher.
I would keep it the same pay fare.
Of course you can/must ask. Some families do that, other do not.
Allergies? Medical conditions? Pay on the books?