@dotbox By default, Opera installs for the current user only. When Opera is installed this way, it won't show in Default Programs as a registered program. But, you'll still be able to set Opera for individual files types and protocols. When you launch the installer, click the subtle, blue "options" link. There you will see the "Make Opera the default browser" checkbox. It should show even for the default current-user type install. But again, Opera still won't show in "Default Programs" that way.
But, forget that and uninstall Opera. Then, you need to install Opera the classic way like most programs are installed. Launch the Opera installer, click the subtle "options" link, set "install path" to "C:\Program Files\Opera", set "install for" to "all users", check "Make Opera the default browser", and uncheck "import data from default browser". Then, not only will Opera be set as the default browser for all users, it'll show up in Default Programs in Windows as a registered program so that you can set all of Opera's defaults easily. That should fix things for you unless there's something else wrong on your system.