1. Take your kids to school on a bike?
If you're living in a dense metro area, your school in within walking distance. So yes. You could bike there, but more likely you'll walk with them. If they go to a further away school, they'll take the subway or a bus.
2. Ride in a business suit?
Yes, I see it all the time with Citibikes. Even on Wall Street.
3. Go to Ikea or Costco?
Sure you'll go there, but you'll have your items delivered like any sane person who doesn't live in BFE.
4. Going to ride in from the suburbs?
People do it every day. You'd be surprised how many showers and locker rooms are right next to corporate bike rooms.
Just because you can't imagine something doesn't mean it isn't normal in another part of the country.