Im almost convinced its because of the aiport extreme.
Unfortunately, the key issue here may be that the AirPort Extreme is not on the Xbox list of "compatible" routers.
I know of no way to change that fact.
We have an aiport extreme which acts as the router and a netgear which is the modem.
You do not have a Netgear modem, you have a Netgear modem/router or gateway. That is why the AirPort must be in Bridge Mode. In Bridge Mode, the AirPort is not acting as a router, it simply passes through the network information from the Netgear gateway.
If you want the AirPort to act as the main router for the network, then the Netgear device must act as a pure, Bridge Mode modem. Netgear support may be able to tell you if it is possible to configure the device that way.
But, even if it is possible to configure the Netgear this way and the AirPort as the main router, the fact still remains that the AirPort is not listed as being "compatible" with Xbox, so there are no guarantees that this will help.
Ironically, when the AirPort Extreme is in Bridge Mode, it is not acting as a router, so the difficulties that you are having are more likely Netgear and Xbox related.
If you want to explore further, take the AirPort Extreme out of the network completely and use the Netgear as a test to see if it will allow the Xbox to function correctly on your network for a few days.
If you have the same problems....as I suspect you will...it is a Netgear problem with your Xbox setup. When you have that problem ironed out, you should be able to connect the AirPort Extreme in Bridge Mode and everything should work well.