If press RMB on the "Edit..." button on the game and go to the "Media" tab, then there will be "Tile", "Background" and "Icon". Can these pictures be obtained as files?

Yes, that's possible. Here are the steps to obtain the files:

Go to gogdb.org and search for your game
Note or copy the game id as shown on screenshot 1 for the example Cat Quest
Copy and paste the following link into your browser: https://api.gog.com/v2/games/
Add the game id to that link. Here for the example Cat Quest: https://api.gog.com/v2/games/1687656736
You should now see a page that looks like the one on screenshot 2. On that page are the links you're looking for. Just click them and download the images.

What about games that aren't on GOG.com but have pictures in the launcher (that aren't on Steam)? For example Boogeyman or Serious Sam: The Random Encounter

the id number is how games are filed... these games background images are downloaded to your local when you mark them as owned so that the game can be seen in the ui

Thanks!