If you receive an eventual approval of H1B, provided you do, your family members will be eligible for H1B dependents status and may be able to attend school. In terms of "Private or public school or college?" you are putting the cart before the horse. First make sure you receive H1B.
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Yes, until reaching age 21, on an H-4 visa as your dependent.
Same off daughter.
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If you obtain the H-1B your children will be dependents on H-4 status. H4 allows dependents to study in the US. They will have to apply for admission to the school of course. That is a separate issue from whether are permitted to study based on immigration status.
Due to the nature of this forum, I often do not have all the information required to provide legal advice. Accordingly, my responses on Avvo are intended as general and not legal advice.