How to reconnect to Airport Extreme hard drive without rebooting

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.

No, this isn't possible. According to Apple Support:

Time Machine can’t back up to an external disk connected to an AirPort Extreme, or to an iPod, iDisk, or disk formatted for Windows.

However, the same KB also notes that:

You can use Time Machine with a Time Capsule, and with USB, FireWire, and Thunderbolt disks. The backup disk can be directly connected to your computer or be on a network. If the backup disk has been divided into partitions, you can use one of the partitions.

You just can't have that drive hooked up to the Airport Extreme.