Last night, after talking with a Mac Genius about a different issue with my phone, I decided to back up my iPad to iTunes (as opposed to iCloud), restore it via iTunes (as opposed to over the air) and restore my backup. She said that upgrading and restoring backups via iTunes is more reliable than OTA updates and iCloud backups. When I had upgraded to iOS 8 and then 8.0.1 and 8.0.2 on this iPad, I had done it all over the air.
I'm happy to report that as of now, approximately 18 hours after doing the update, this issue and my issue with spotty Wi-Fi in my house have both gone away, at least for now, with heavy usage.
If the problem comes back, I will let everyone know on this thread. If you don't hear from me again, assume that this fixed the problem and I would recommend everyone do the same.
Good luck!
-Scott