Have you considered the AMD APU? I'm pretty sure one of the larger on-board IGPs (384SP or 512SP variety combined with 2133MHz memory) can handle your needs. CPU portion is a lot faster then any Atom.
It'll reduce complexity since you can loose the discrete graphics card.
While these APUs are pretty impressive, we really need all the power of GTX 970 level GPU. I don't know if it makes sense, at least cost-wise, to use an APU then instead of a regular CPU.
That board you linked to shows a nice, short APU cooler. We want to make the setup as short as possivle (that's where the idea of positioning both the motherboard and GPU horizontally and connecting them with a extender cable comes from). So if there are CPU coolers like that I'd like to know, as well as what CPUs will work with them and how to prevent overheating. This is a question deserving its own thread I know, but this thread explains why I would ever want to get a tiny CPU cooler.
Did you check out that chassis I linked to earlier in the thread? It does precisely that and pretty much everything else you want while being remarkably small.
There is little reason to DIY a chassis, unless its for a very specific purpose.
Unfortunately it is for a very specific purpose.
The case for the device has to be cylindrical with very specific dimensions, with two floors, first containing the motherboard with the GPU and second floor a special DC motor, small specialized video projector, some optics, mechanical parts and a microcontroller.
So yeah, the case has to be custom built.
A spiritual successor to this device:
Goal is to build it as small as possible.