Yes the cord is fully adjustable

Are there instructions as to how to shorten it?

You should be able to loosen up the screw holding the cord to the ceiling canopy and then pull the cord through the canopy to the desired length. Then just tighten the screw and cut the extra cord not needed.

There's a plastic part that appears to be wedging the cord into the fixture to secure it. Any clue how to get that thing out and how to replace that part?

just use a regular screwdriver - that plastic part is a plastic screw

I'll try it when i get home but that plastic part was bell shaped.