How can I replace this unusual kitchen sink drain pipe?
very easy fix,    just cut the pip from the water under kitchen sink, start fresh.
remove all drains,
you need:  1 two part waste      1 trap kit   1 no-hup 1"1/2 to 1"1/2 tuppler and ask for an under sink AC adopter to connect the hose to your garbage disposal to.

very easy, but maybe YouTube might have some photo to show you.
also i advise you to use brass parts, and make sure you have a copper cutter big enough to cut pipes to size, but you will new a saw to cut drain under sink. 

i hope this helps :)