If it's "very old" it's probably not plastic which has been prevalent in the plumbing industry for the past 20 years or so.

It's probably POT metal.

Pedantic comment follows -

This is NOT dry rot, this is corrosion. Rubber dry rots, metal corrodes.

For us in exposed locations only in my opinion.

"Artsy" pictures taken at weird angles suck.

Tubing Cutter

Run a dowel through the tube leaving enough to hold on to, then use a tubing cutter.

The manufacturer says:

Ensure all surfaces are clean, dry, and free from any previous finish. Sand the substrate with a fine 150 grade sanding paper or 150 grit sanding disk, removing all traces of dust.

They know their product - not sure why you'd do anything but what they recommend.

If it ain't broke, don't fix it.

The problem is that waste piping must be installed so that the flow is directed. You have a san tee laying on its side with the outlet dropping out the bottom. With a significant flow the water from the sink on the left will by-pass the outlet and flow directly into the disposer (or vice versa). It's a code violation not to mention a clog waiting to happen.

The dishwasher airgap discharging into its own trap is suspect for the simple reason that traps should only be installed on fixtures that will receive regular discharge. It is assumed that a fixture with a faucet will receive water from the faucet on a regular basis but that is not true of a dishwasher. For example, my mother has never used her dishwasher in the 5 years she's been in her apartment - her trap would have lost its seal and sewer gas would be entering via the dishwasher. You may use your dishwasher everyday and never have a problem but in many jurisdictions this is a code violation.

Both issues are relatively easy to fix.

I don't know anything about your electrical question but the plumbing for the sink and disposal is totally screwed up.

So you know all the codes all over the world?

Blanket statements like "there's no code" should never be made (or believed) on the internet.