A "double NAT" condition is indicative of having two routers with NAT enabled in series. It appears that your D-Link modem is actually a combination ADSL modem & router. Typically, in this configuration, you would want the downstream router (in this case the Extreme) in bridge mode.
Once in bridge mode, the Extreme should NOT lose Internet connectivity periodically. If this still continues to be the case, I would suggest a complete power recycle of your networking equipment, and then, reconfiguring your Extreme as a bridge.
Please post back after doing so to let us know your results.
In addition to the expert advice from Tesserax, in order to configure the AirPort Extreme in Bridge Mode to work correctly with your D-Link modem/router.....
Open Macintosh HD > Applications > Utilitiies > AirPort Utility
Click on the AirPort Extreme icon, then click Edit
Click the Network tab at the top of the window
Make sure that the setting for Router Mode is set to Off (Bridge Mode)
Click Update to save the changes and allow 30 seconds for the AirPort to restart
Then, power cycle the entire network by powering all devices off
Wait a few minutes, then start the modem/router first and let it run a minute
Start the Airport Extreme next the same way
Continue starting devices one at a time until the network is completely powered back up