22 * 22 * (24 (in^3)) = 50.2857143 US gallons.
With regards to cube tank popularity, they are really popular in smaller sizes, such as the cube gardens that ADA and various other places sell.
On hoods: For the hood, you could simply make a cover out of acrylic, plexi, lexan, or even glass. 2ftx2ft is a very common size for plastic sheets to be sold in. Make cutouts in the top where you need to have space for filter inflow and outflow, as well as any other equipment you may have in the tank, and you're done. A hood or cover is not really necessary, it just depends on the type of fish you'd like to keep. Personally, I prefer having an open top, but that is just my opinion of couse.
On lighting: You may want to go with 2 different 24" fixtures. Since your tank is quite wide from front to back, having the lights staggered on top of the aquarium with a gap of several inches in between each fixture would help assure that light gets distribuited through the entire tank. I'm a noobie myself though, so I could be wrong on this.