The "error occurred while checking updates" message will happen if autoupdate.geo.opera.com is blocked in your country. It's definitely going to happen if you renamed/deleted opera_autoupdate.exe. So, those are expected and you'll have to update manually with the offline installer each time.
As for the red update notification despite you having the latest version, I'm not sure why it's there and how to get rid of it. You could try creating the OPERA_AUTOUPDATE_DISABLED environment variable for your Windows user as mentioned here and restart Opera (or log out and back in and then restart Opera). You can also always launch Opera with the -disable-update command-line switch mentioned in the same post to see if that helps.
But, the notification might still remain. It might default to always showing in the face of some update error. Not sure. The red notificaiton might come from checking some other Opera site that tries to do an extension update check or some of Opera's components at opera://components. Or, it could be something else that Opera checks.
You could do a test though. Goto the URL opera://about and take note of the "profile" path. Then, close Opera and rename the profile folder. When Opera starts up and creates a new profile, if the red notification goes away, then you know there's some stuck setting or something in your profile. You can then close Opera, delete the new profile, and rename your old one back, and then investigate the files. Might be something in the "Preferences" or "Local State" or "Secure Preferences" files. Those are best views with JSONedit. The "Secure Preferences" one you won't be able to edit though as it's protected and your changes will be reverted probably.
If it still happens with a fresh profile, I would do a thorough uninstall of Opera and reinstall (except do not delete your profile and cache folders) to see if it helps.
In Opera's profile folder, there's an autoupdate log you can look at. It might always contain some "resource_l10n_handler.cc" and "protocol_impl.cc" errors. But, there might be some other errors in there that will shine light on what Opera's trying to update.
Or, if you could temporarily use a system-wide VPN on Windows that works in your country, and enable Opera's updater and check for updates, the red notification might go away.