A recommended course of action would be to backup your data onto a different (external) drive, wipe your Mac clean and start over. This could be the most sure-shot way to ensure that your computer is free of any/all malicious apps/permissions/plug-ins.
When backing up the data, make sure you backup all the crucial data and don't miss anything. Files can be ...
I personally recommend Malware Bytes: https://www.malwarebytes.com/mac/
You do not need the paid version, you don't even need the trial of the paid version. Download the software and decline the trial. You just want to run a scan of your computer. It will give you instructions to resolve any issues, in most cases automatically.
Ordinarily you should still be safe after just mounting the DMG - as long as you haven't actual run anything from within the DMG by double-clicking an icon or similar.
So normally no, you wouldn't have installed the Skype file just by mounting the DMG.
By looking at your screenshots you seem to be running a modern version of macOS. Are you instead running ...