I found the fix for my issue (please read on)... My issue was similar and began as soon as I upgraded my iPhone 4 to iOS 7 from iOS 5. Also, my wife bought me an iPad Air, and bam - same issues with it right from the start. Keep in mind that my wife's iPad 1 and iPhone 4 on iOS 5 continued to connect to our home wifi network and the App Store without issues.
The problem: when browsing the App Store, my wifi connection would hang. It would also hang anytime I was downloading apps. Safari would hang as well when I was in the App Store, or downloading apps. The only way to get anything to 'unhang' (and only temporarily) was to turn airplane mode on and back off, and even then it would hang again within 10 seconds or so and I'd have to do it again, and again, and again. FRUSTRATING. It was taking up to 20 minutes to download an app. If I wasn't downloading an app or surfing the App Store, my wifi connection was good most of the time. Still, some apps like Flipboard would hang even when I wasn't in the App Store, until I turned airplane mode on and back off and I'd have to continue doing this as long as I used the app - again, and again, and again....
To troubleshoot: I tried pretty much everything suggested in the Apple forums, except the refrigerator trick (no way was I falling for this;). I reset my device network settings and turned my device off and on several times. I turned any and all settings you can think of off and on again, and again, and again. I even did 2 complete restores of my iPad Air. I also went through every single setting in my router (D-Link DIR-615 - a/b/n compatible) and modified the settings per Apple's wifi setting recommendations, and per suggestions here, and per many other message boards -without success. I thought maybe it was my router's DNS settings (openDNS), so I tried several other public DNS servers - again, without success. I took my iPad Air to work to see if it would work there. It did - just fine.
The solution: Finally, just about ready to return my iPad Air, I decided to try a new router as a last resort. I had read something suggesting that a dual band - 2.4 / 5 Ghz - router, with ac might help. It did. As soon as I hooked up my new Netgear router and got wifi up, apps instantly downloaded, and no more hanging.
So, I'm not happy that I had to upgrade my hardware to resolve the iOS 7 issue, but it was probably time anyway. I hope this helps someone else and saves them the hours and hours I spent working on this....