i hate to burst your bubble but Its been thought of and done before, its called a RAGB (right angle gear box). This is how some riding lawmowers work, a peerless 700 transmission is used in many rear engine riding mowers and is a simple right angle gear box with 5 different speeds. The reason people dont use vertical engines is not because its hard to do but because it takes up twice the room of a horizontal engine and is just not worth it, for all the space they take up you could run a small sled engine.
Personally I like the right angle belt.
2 or 4 stroke? industrial or performance? you cant determine HP by CC, for example my 100cc 4 stroke dirt bike puts out around 8hp max, where as a 5.5hp tecumseh is 200cc.
fill it up half way with 75-90w gear oil, and tap a hole for a breather, put in a fitting and attach a hose to that. Also a few heavy duty straps around that case wouldnt hurt, i doubt it could handle the stress of a kart without the cases splitting. Are those gears steel or aluminum? if there aluminum forget it, if there steel then strap the case fill it with oil and hope it doesnt split.
What size engine do you plan to use?
Yes you can cut off the axle and weld a sprocket, but you may want to add a pillow block to save that bushing.
I dont understand what you mean by "run it sideways"