I find the limit on thin to be determined by the height of your panel and the angle you want it set back at. If you're going for a significant layback on a vertical screen like a Galaga, that'll necessitate a back wall significantly deeper than a more upright mounting and/or a shorter screen will.
In my case, Mimic also had the extra invisible height of the hidden screen area to contend with, there, so I could only lay my screen back to about 15 degrees before it was getting too thick overall for my preferences.
Footprint wise, I'm at about 13" deep. While I did play it freestanding for a couple weeks, I sure wouldn't recommend it or let a random stranger do it - I had to supervise my friends, and had to catch it and stand it back up a few times when they got exuberant with a both-sticks-back Battlezone dodge. It was designed to be screwed to a wall, and pretty much has to be given where the center of gravity ended up.
So, if you want to go thin + free-standing, I'd say you probably have to go deeper than 13", or get the CG further back than I did, or both.