University and Bellevue are affiliated. I always think of them as the same hospital although housed in different buildings.
Everything I said still holds. Just because a doctor or nurse is an employee of one hospital does not mean he/she is allowed to work in all other affiliated hospitals. I deal with this issue constantly in my current job (a different health care system). Credentials in one institution do not automatically transfer to all affiliated institutions. It also does not mean that insurance or electronic medical records transfer. All the other points would obviously still apply.