In the end....do what works for you, I'm just tossing out ideas....
...but you do want them on the floor without the large space underneath.
You can make them free standing by simply adding a piece of wood at each end to the bottom of the frame...perpendicular, and so that you get about 6-8" sticking out front and back....that way the gobo sits flat on the ground.
The other option, with no legs of any kind, is to hinge them together, but so that the hinges are the kind where you can just lift one of the frames up off the hinge to easily separate them. When they are hinged, just set them with a small angle, and they will stand. In most recording situations, you are probably going to want to do that anyway with the two frames. You can even include the kind of legs I mentioned above, so when if/when you separate them, they can still stand individually....and/or you can put those legs on one end, but make it so the hinges work with the frames flipped, and the legs end up on top, with only a single screw/pivot point...and then you just twist/turn the legs out of the way, on top.
Oh...AFA the back...I would definitely add at least a 1/4" piece of plywood, or even better, 1/2"...because for gobo use, you are looking for some isolation.
With the gobos open/soft on both sides...you wont get much of that. just a dulling of the sound that passes through.
With one hard side and one soft, you can choose how to place them...isolate what's coming in or going out.
Plus...you will at the same time, cover up that rough looking backside...so they will be 100% finished looking all around.
Check some pro gobos...they almost always have one hard surface side. If you are worried about reflections off the hard side, you can add some cloth or a piece of indoor/outdoor carpet....etc.
Anyway...you do nice work.