Do a test. Goto the URL opera://about and take note of the "profile" and "cache" paths. Then, close Opera and rename those folders. Then, use Opera (without Opera Sync) for a bit and test things out. Log into sites, close Opera and start Opera again to see how things work.
If things work fine there, something in your profile is messed up. When you're done testing, if you don't want to continue using the new profile, while Opera is closed, delete the new profile and cache folders Opera created and rename your old profile and cache folders back to restore things to how they were. Then, you can investigate further or export and back things up, delete your profile, start with a new profile and import the things you exported.
It also wouldn't hurt to restart your computer and uninstall and reinstall Opera to make sure Opera's program files are in tact. (Don't choose to delete your data when uninstalling.)
