Well tried several things. And it definatly is Avast that is causing Opera to close.
I checked every available log file, but there was nothing in them related to this.
No minidump was created (as could be expected).

Enabling ‘show detailed info’ showed me it always happens when scanning the Opera cache folder.

Disabling standard-shield stopped the closings.
Adding Opera’s cache folder to the exclusion list also stopped the closings.