We have recently had numerous flood map changes in our lending area. Once notified by our flood determination service provider that a specific property is now in a flood zone, how much time do we have then to notify our customer and request for flood insurance? We have had numerous properties affected by the map changes, so this is more than just one or two flood letters that have to be sent out.

The only guidance would be "promptly". Just make sure your definition matches the expectations of your regulator. A week might be able to be justified depending on how many you are talking about. A month, not so much.

What if a property is mapped into a different zone, but is still not in a flood zone and does not require flood insurance? Are you required to send notice of that to the borrower?

No.