Download Opera 12, run the installer, click "options", set the "install path" to your existing Opera so the button at the bottom says "upgrade", check "use Opera as my default browser" and install. That should fix any registry keys for Opera 12.

Then, if you have to, go into default programs and set Opera as the default again. Also, goto "Alt + p -> advanced -> programs" and check the box to make Opera the default. Also, goto "details" and if it doesn't load default program and gives you a list of stuff to check, check them.

If you have to, check that the command in "HKEY_LOCAL_MACHINE\SOFTWARE\Classes\http\shell\open\command" is correct. Check the ones in HKEY_CURRENT_USER and HKEY_CLASSES_ROOT too. Also, check the ones for https.