would a faulty secondary air pump cause my car to shut off while idling ? I have a 1999 BMW 323i

Depends on how the pump fails if it is constantly running they yes it could after the vehicles goes into closed loop. But if it has just failed then no it wouldn't and you need to look else where. But neither of these concerns would cause a hard start. Many things could be causing the issue your having with it shutting off at lights. I would start with the basics unmetered air leaks. Good luck and hope this helps.