Download Opera. Uninstall Opera (don't choose to delete your data if you don't want it deleted), delete "C:\Program Files\Opera" if it's still there, launch the installer, click "options", set "install path" to "C:\Program Files\Opera", set "install for" to "all users", uncheck "import bookmarks and data from default browser", check "Make Opera the default browser" and install.
That's how you should install Opera. And, that makes Opera available in Default Apps/Programs.
In the registry, there will be an "HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\OperaStable" key. In it, under the Capabilities key, will be the FileAssociations and UrlAssociation keys that specify the file types and protocols that Opera supports as being the default for. And, in "HKEY_LOCAL_MACHINE\SOFTWARE\RegisteredApplications", there will be an "Opera Stable" string that points to that Capabilities key. And, in "HKEY_LOCAL_MACHINE\SOFTWARE\Classes", "HKEY_CURRENT_USER\SOFTWARE\Classes" and "HKEY_CLASSES_ROOT", there will be an "OperaStable" key specifying the command to Opera's launcher.exe. All that must be right for Windows to show Opera in Default Programs.
You should run regedit and make sure those keys are there after installing Opera. And, make sure the path to Opera's launcher.exe is correct in each shell/open/command key for those.
