Some more details to the still unsolved issue of downloading apps with Google Play Store:
It also happens with some (but not all) apps at the APKPure store, which are shown as to be updated. Downloads also don't work with the XPosed Installer, same behavior as with the Play Store. At the F-Droid store I experienced no such problems.
It looks as if the issue has to do with the download process from the Play Store server. The Play Store app is telling, that it does download (an app) and shows a green line, running from left to right, but there is no download icon at the status bar. After some time this process stops an I get an error message , "errorcode 495". At the Google Help Desk they specify six steps to eliminate the error, I tried it all without success.
Does anybody know a tool, which analyses the download process and does it show at a log file?