Thank you for contact about your concerns especially these difficult crisis days.
I'd like to inform you about the aspects that you mentioned and I will try to answer all of your worries.
The compatibility list of the operating systems exists in the download page at the bottom right place (please see the screenshot).
If you are using a free version and not licensed, then you must use the last version of the TeamViewer app. Licensed users have many extra features enabled and one of them is to connect between different versions of the app.
If you want to take a look at this related post especialy to the answers of Esther and Youri because they have the full knowledge of the products.
So in the case of a free license, it is logical that your computer must be using the latest version. If it is not compatible with the latest version, then you must change something, either to update the macOS (if it can be updated), either you can buy a license.
The version that does not needs to be installed is the Quick Support app and it is for accepting incoming connections to the computer. You can download it from the website, run the .DMG file and you will see the following screen:
Just double click on the blue icon and it will run the app instead like the following screenshot: