Update: my problem is solved.

The support team asked me to do a complete uninstall, manually clean/delete some files, make sure the Remote Desktop Services were started up and then re-install. This solved the issue.

Maybe it can help some others with similar issues as well…

From the support team:
[i]May I ask you to reinstall Avast on your computer once more following the steps on our official webpage below?

https://www.avast.com/faq.php?article=AVKB94

Once you will uninstall Avast and remove all the listed files manually, please verify the status of the service “Remote Desktop Services” again, so that its “Startup type” would be set to “Automatic” and so that the service would be properly running.[/i]