Is there anyway to restore this Airport device? I tried swapping cables, and booting the router and the IP's "RADIO")
There are at least three reset methods you can choose from. Please check out this Apple Support article for the details.
This is the third Apple router that seems to have had the same problem. I have also had the problem with much cheaper Netgear routers.
Our internet provider (a wireless provider) said that a stray bit of lightning probably got to it.
Does this sound reasonable?
Actually, yes it does. You may want to consult with an electrician to see if your home's electrical system is properly grounded if lightning storms is a common occurance there. Other than that isolating the Extreme with a dedicated good quality surge protector may offer some protection.
Can you recommend a good quality surge protector for ethernet lines?
There are a number of vendors out there, but I have been using Tripp Lite for a number of years and have been happy with their products. The do provide surge protection models that include Ethernet. You can find them here.
Are the reset methods methods that you refer to likely to restore lightening damage?
Unfortunately no method of reset will actually restore a device from any electrical surge damage regardless of the cause. I would recommend that you try the "factory default" option as it is intended to restore the AirPort base to its "out of the box" configuration. You may want to try this several time just to make sure it "sticks."
Happy 4th to you as well!