it is possible as long as your not using the quad axle its self. they taper off from the center bearing carrier to the hubs. its also something i never tried so maybe someone has an idea if your planning on using the quad axle.
you should be able to then. do you plan on welding the sprocket on or cutting a key into the axle so you can easily play with your gearing.
you shouldn't weld you sprocket on.
Yep you can move the sprocket to another part of the axle, the current sprocket carrier would most likely be keyed onto the axle, unless the axle has a key way along it ( which i doubt it would) then you will need to cut a key way into the axle where you would like the sprocket and this would help you when lining up the engine.
I would stay away from welding the sprocket straight onto the axle it would be hard to get it straight and obviously there would be no movement for lining up purposes
Never, ever, ever, ev... Well- you get the picture. Do not weld anything to a shaft.
If you can find/make/rig a hub that will fit, then yes.
Or, you could use a jackshaft or shaft extension to relocate the chain run to where you want it.