Version String in FamilyName (or filename)

Is it possible somehow to add this automatically to either the familyName and/or the fileName (for instance when adding those in a Custom parameter)

Best solution for this would probably be your own export script. For an example, look at what Nico Hagenburger did in his repository.

Okay, thank you!

Or a script to update the version number and familyName (and/or fileName) together and use that instead of manually updating the version number.

You could write a Filter plugin that does just this.