He might be referring to the nipple on the disposer, where the DW drain is connected. That nipple does tend to be the first thing to "rot away" on a G/D.
All of the hand-held showers that I'm aware of screw directly on to the old shower arm. I don't know why you would need or want to replace the shower arm.
Some fittings require more teflon tape than others. Also, a quarter turn with a channel locks past hand-tightened is often needed to get a good connection. Just don't crack anything.
It would be better to use compression fittings, so you can turn them tighter if you have to. With this type of...
Last year, when I visited my parents in SE Michigan, they got their main line rodded out for $80.00 from a reputable plumbing company. So, prices do vary tremendously.
In my research, the only 3-way ball valve I could find with a plastic tube coming down is for winterizing hot water tanks in RV's. The tube somehow sucks antifreeze into the tank from a bucket. But my guess is, you don't live in an RV. :)