How often do they check vision in public schools?

Depends on district. Most check the vision on entry. A few do it every several years. If you have concerns about your child's vision, you should see a pediatric ophthalmologist and get it checked out.
Depends. This depends on the school district, but many check once a year. Your child's vision will also be checked yearly at their annual physical.
Maybe every 3 years. Public schools do vision screening on the students in elementary and middle school every few years (the exact number probably varies with school district philosophies and budgets). However, if parents have vision problems such as lazy eye, strabismus, nearsightedness, or farsightedness, then they should take their child in for an eye doctor's exam before first grade, even if the child seems fine.