As I wrote, associations didn’t matter. I just opened the cmd.exe console and wrote something like:
"c:Program FilesEmEditorEmEditor.exe" "c:Program FilesEmEditortemplate.xml"
and it did not work.
But I found what the source of the problem was: the EmEditor instance that was running on my machine was elevated because I didn’t untick “Launch EmEditor” checkbox on the installer’s last page, and it ran EmEditor with the current installer’s privileges. Later when I was reproducing the issue, I started EmEditor from PowerPro. PowerPro itself is running elevated, but I set the “Run as normal user” checkbox in it. However, there seems to be a problem in PowerPro itself and this option does not work: EmEditor starts elevated despite of this option. I did not notice this at first, so I thought it was something wrong with EmEditor.
Now I tried starting EE directly from Explorer, and when it runs with normal privilages, opening files from outside works correctly.
Sorry for the false alarm.