I really love my current setup. I use Zorin OS Lite 16.1 and it works great for playing old games.
Whenever I place a shortcut on the desktop, it’s icon is showing properly, but if I create a folder on my desktop and put the shortcut inside the icon goes away.
I don't think it's possible what you're trying to archive here, as Desktop is a "special" place where .desktop files shows its icons. If .desktop files are placed anywhere than the Desktop it will show as a normal file.
May I suggest you use Plank for your games. You can set it to auto-hide and place it anywhere at the four corners on your screen. Plank is really lightweight and won't take much to run.
I like to keep things organized but my five year old son relies on the icons to know which game to start. Plank seems like an even better option than my folders.
You can drag the desktop icons into Plank. Right click on Plank to get its preferences (it may be a bit difficult to hit it).
If you nee help with Plank, let me know.