Download the offline installer (the 64-bit one if you're using Windows 7 64-bit) from https://www.opera.com/download, close Opera, and uninstall Opera (don't choose to delete your data when uninstalling). Then, launch the Opera installer, click "options", set "install path" to "C:\Program Files\Opera", set "install for" to "all users", check "Make Opera the default browser", uncheck "import data from default browser", and install.
Then, Opera should stop bugging you and Opera will appear in Default Programs.