AirPort Extreme is disconnected from the internet when in bridge mode

Contact your upstream internet provider (Cablevision) and let them know you wish / need to manage the network and want their device to operate in bridge mode. Often someone will post the details on how to get into your modem and re-configure it in cases where the provider doesn't want to assist you, but I've always had good luck asking (especially the techs that will have to come to your location for trouble calls.) If you call once or twice and explain you call each time this Double NAT error happens, they will quickly realize what you need to run your network without having to call them on an ongoing basis.

The Airport hardware doesn't have anything to do with this detection, so the software in the router is detecting a change in the upstream network configuration and reporting it.

Usually I've seen this with metro-wide WiFi networks where a fault will change the routing through another router that is misconfigured or not normally serving the modems of customers directly.

Unless you're changing the settings of the Airport, the cause is almost surely up stream from the device itself.

The Amber light could mean many things.

Steps I would take:

Install new firmware http://support.apple.com/kb/DL1481?viewlocale=en_US&locale=en_US

Second:

Factory default reset

Does your modem have a router built into it? If so, you need to put that either into bridge mode or router behind router mode.