Not to throw anyone off their particular trail, but after reading this forum and all of the others that seemed related to this issue in the "AirPort" forums, I finally did resolve my particular issue. However, it was not before I tried every version of the AirPort Extreme/Express firmware and all of the great suggestions herein. I even repaved the entire hard drive and reloaded Snow Leopard from the install disc and did not copy any settings over from my backup (I got the networking running before individually bring all my data back so as not to corrupt the settings with old data). I'm back up now (finally and solidly). In summary, I do not think this is an AirPort issue and I don't believe it's an Apple Networking issue. I don't think you need to wipe out your hard drive either. That's the teaser -- here's the short story and then my solution.
I have an Airport Extreme Gigabit Ethernet router and an AirPort Express configured to extend the network and mostly used for streaming iTunes to a stereo system. The main router is connected to a Qwest Actiontech DSL modem (R1524SU) circa March 2004. After upgrading to Snow Leopard, it all went to He--. I'd be surfing with Safari and the spinning wheel would start but then never connect to the targeted website. It would just keep spinning, iTunes would stop (if streaming music), Mail would stop, the whole thing would go down the tubes. Powering on/off the Airport card in the laptop didn't do anything. Using AirPort Utility to reset the router functioned, but did not clear the issue. The only solution for me was to cycle the power on the DSL modem and then cycle power on the router. Note that this also occurred a couple of times with 10.4.11 and 10.5.8. But, Snow Leopard was a whole new level of pain -- the thing would lock up every 2 minutes!
After about a month of messing around on and off, I finally gave up. But, a night finally did come where I had to get something done and needed what was on the laptop. I turned off the AirPort card, turned off the AirPort Express router (just because I was mad), and plugged an Ethernet cable directly into the DSL modem. Guess what? it locked up in less than 30 seconds -- without the AirPort in the loop. Eureka! I immediately called Qwest and asked for a replacement for the DSL modem (after a long troubleshooting session with a nice boy named "Victor" who's line test told him that my modem was "dropping authentication"). Because I was renting it, they didn't complain and shipped it to me two days later.
The new model is the Actiontec PK5000 and it was simple to set up and hasn't skipped a beat. It has 4 Ethernet ports and built-in wireless, but the administration tools let you turn that off. Other than password there was no setup required. Back on line in 10 minutes. It made me laugh at how much time I wasted on trying to figure out what was "wrong" with the Extreme and Snow Leopard. I think it's as simple as Apple moving the networking software forward with the latest standards and my DSL hardware remaining static without a firmware upgrade. (Don't get me going on how there's no upgrade on the Qwest site for Mac user's.)
The lesson? We spend all that time loading all those Mac OS and software updates on a weekly basis but tend to forget the other important pieces of equipment in the chain.
I hope this helps others with their problem.
Angus