To answer your questions based on my experiences with axe robots:

-Would the diagram drawn out above even be feasible, what would be the best "ratios" and if so, would it increase power, or would it just slow it down too much?
Yes, it is feasible. I do exactly this 2 stage reduction with solid gears in my Middleweight. I would aim, in the FW class, for around possibly 8:1. Azriel (my FW axe robot) uses a single stage 4.5:1 reduction and although the axe is quick it lacks downwards oomph. I'll be addressing that in a rebuild soon enough.

-Where would be the best places to get such sprockets, along with chains, axles, bushings and keys?
I use https://www.gearsandsprockets.co.uk/ for my stuff. They have all you need except keys. Bearingboys.co.uk sell key steel as well as gears et al but you will need to find someone who can machine a key into a shaft. Roboteernat has done mine in the past.

Would it even be possible to fit all those chains and sprockets into a 5cm gap between the uprights?
This I can't answer for you. You need to sit down with a pen, a piece of paper and a calculator to work out how wide each sprocket is based on their engineering diagrams, how wide the chain is and then add them all together to get your answer. When I was working on Jibril, I used CAD and a calculator to work out the overall dimensions.