Sorry! :) The output shaft is the transmission output shaft, which is where the front chain sprocket mounts. It has splines (ridges) that keep the sprocket turning with the shaft rather then just sitting there spinning. The previous owners of my project KDX were not kind people, and ran either the wrong sprocket on there, or ran the right one far longer than they should have. As a result, the splines were just about gone. Nothing to do but replace the transmission output shaft, which means splitting the cases and doing a complete teardown. Yours is probably fine, my was damaged by previous owner stupidity. Given how badly this bike was treated, it was probably the right thing to tear it all the way down regardless...
By jug, I just meant the cylinder, sorry. :) The part that contains the piston.
If you can drag a fingernail around the inside of the cylinder and feel it click over grooves, then you probably need it redone. The service manual also gives wear limits for size, but it can be tricky to measure.
I think stock the cylinder is a tungsten vapor deposition coating of some sort... not the best, but not awful. So if it's in good shape I would keep running it, but if it's grooved you want to re-do it.
For mine, I went with the Eric Gorr 225 cc rebore and Nikasil plating. Nikasil is great stuff, transfers heat like crazy (to get the heat out of the motor) and lasts forever. It's the right "high tech" solution. Not cheap, but not awful either given how long it lasts. I really thought that was the only real option when I did mine. Going to 225 means its a standard Wiseco forged piston size. Out the door, I think the whole thing (bore, plate, KIPS cleaning and timing, Wiseco piston kit with bearings rings and clips) was $500 or so.
When we bought that 220 for a buddy, he said he had a local guy with a good reputation just slap in an old school iron sleeve for under $100 (including labor). Not as high tech, but it seemed to be working really well. Not sure what that really costs, but it's a straightforward job. And if you get damage on that, you can find locals that'll rebore iron cylinders for $60 or so, and just put in an oversize piston.
So I like my fancy Nikasil setup, but there are other decent options.