Your comments

Could you elaborate on this, please? Origami is about custom dielines - you create them in Illustrator and it then folds them up for you. What exactly feature you miss?
So you want to be able to replace all the images at once? Like having a list of currently used images and the ability to modify that list, automatically replacing the assigned textures?
We've got tint for that. Doesn't it work for you?
We'll think about this, thank you. So far the suggested option is to export shapes to Boxshot or any other editor for further composing.
What if you export the shape to Boxshot/Owlet/any other 3D editor and then rotate and render it there? That's the best way of doing this so far.
I feel your pain, but so far we see no way of simple and effective way of doing that. Suggestions are welcome.

We can't mark bottom with an edge, as shape may rotate around the edge, so the angle to the floor cannot be defined that way.
Step and repeat should help for that particular project, unless all the shapes are different in details.

Here I made a simple card, then multiplied it with "step and repeat", then cloned the group to make a second row, and finally made the layout like yours:
Well, it works a slightly different way, but at least you can distribute a single object around the scene and then tune them up one by one.

Anyway, I got your pain, will have a look at what we can do about that.