Often enough that high house prices can keep your kids away from most bad kids. I think it's a crude mechanism that still allows too many bad kids in and keeps a lot of good kids out. I don't like it. It's also the best available mechanism in many countries.

Probably not in desire/at heart. Constraints on time can be real though, and can impact child economic outcomes.

IME yes, upper middle class people are much less likely to let TV/video games raise their kids, are much less likely to be divorced/have multiple families, etc. They tend to be very involved parents. Being conscientious, agreeable, and intelligent tends to make you a good parent and bring economic success. Being impulsive, antisocial, and stupid tends to make you a bad parent and not bring very much economic success. So you obviously end up with a correlation.