How long does it take for his permanent tooth to come in after his baby tooth falls out?

Just depends. Some teeth come in before the baby tooth falls out and others can take a month. You should see a dentist to make sure the adult tooth is there and is erupting properly.
Usually quickly. It depends on the situation. Some teeth are lost and the permanent one doesn't appear for years, usually it is right there as soon as the baby tooth falls out. A check up with an orthodontist or pediatric dentist can easily identify if there is a problem.