I do not understand the question, but regardless of my lack of understanding on what you need, the answer is mostly yes.
I wonder if it's technically possible to generate fake 3D sprites
Most 2D image manipulation programs have "skew" and "rotate" You can do that on bitmaps even on vector-based programs.
Most of the raster manipulating programs have some kind of perspective deformation on raster images.
with sharp pixels
Turn off anti aliasing.
by using only pixel information from 4 flat images and manipulate them to set proper object dimensions with different aspects of depth (see last image)
Yes, but no program will make it on its own, you need to tell it what you need, what do you need as top image, which deformation you need, which perspective.
You could save one macro or use linked textures. That is how most mock-ups work.
so the end result is a fake 3D object
No. You will not have a fake 3D object, you will have a resulting image that your brain will interpret if done correctly as a 3D object. Cartoonish if you need. But will only be planes deformed and put together to be interpreted as a 3D object.
that can be rotated precisely to many angles within a given limit.
So... What you need is... an actual 3D object that can be rotated to some specific angles.
Get some 3D programs and assign the textures to the faces. Then define your lens FOV and angle, and render settings.
Some 2D image programs have some 3D capabilities. Illustrator, Corel Draw, Photoshop. Even a modern web browser and CSS3 can do it. Some just use the tools I mentioned first, but some do actual 3D operations considering the point of view.
So the answer is yes.