Yes they do, try to contact them in advance.
They have two options one is take the children to a deluxe resort like the Polynesian and they will watch them like a daycare school would. The more expensive way is the second and a babysitter will come to your room and watch the child there. If it was me I'd go option 1 unless he is very shy around people he doesn't know
Yes, you have to contact them
The hotel itself does not but disney does have a daycare/sitting service located at the more expensive resorts
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