I agree with E. in all his thoughts, but I think there is a way of doing what you want perhaps. Not a project for the faint of heart though and you will have to be prepared for it to perhaps not work.
On the subject of a graphics card in the build, and wanting good performance from it you really would be squeezing the Cube format to the limit.
I don't know if you have been reading my thread where I am experimenting with getting a water cooling loop in the Cube? Well, to be honest I think that if you are seriously trying to get a GPU in there then maybe cutting the backside of the Cube and getting rid of the handle mechanism to make room for serious airflow and extra component space would be a requirement to avoid cooking. So you could borrow that idea from me with pleasure.
My build will I believe show that it is feasible to draw good amounts of air in from the bottom AND the top and eject warm air out of the back with a good sized fan combination - mine is of course 2 x 120 in push pull on a radiator, but you could get 1 x 140 even I think (need to be careful with the metal and perspex cutting though!) and if you were air cooled then instead of water then you can potentially free up - without there being any handle or heatsink in the way - a good amount of space. BUT it will be a major re-engineering project.
Without that kind of major re-work I think that the GPU and parts needed to drive it will cut down airflow to almost nothing in a conventional bottom to top system and you will fry (just my opinion).
The other question is power. Something like my Zotac board will run a good graphics card as it has a proper PCI-E 3.0 slot. It will fit in the Cube, but think about the PSU requirements. With a i7 3700k I know I can run an overclock with a 160W pico....and stay within the PSU limits. Add a GPU though and it is a different story. My GTX 660 flat out requires 150W on its own (and no that is not going in my build). You can of course use a less hungry CPU but that will not gain you enough power headroom to run a graphics card to full potential still.
You can get a in car PSU that is a bit bigger than the Pico which is capable of 250W but you'd be hard pressed to find a mains adaptor that would work that. Highest power mains adaptor I have seem is actually the Cube one which is supposed to be 205W and is coincidentally compatible with the in car PSU board (that takes up to a 30v input). I must admit that is a combo I am interested in trying in a future build.
You could in theory squeeze in a more conventional supply but that is also a heat generator and takes up space (and removes space for cooling...).
Another possibility I thought of which I have not researched (as it is too extreme for me) would be to have a regular Pico PSU for your system and then add a separate input for a 12v line of a 150W mains adaptor to exclusively plug into your GPU......
So, technically? Yes. Practically - possibly? Hard work and in the end maybe not a great idea - ......?
It would be fun to watch.
PS excuse the stream of consciousness reply here - just the way my brain works as I type this!