I am trying to export a docker container that uses a mounted local volume as its root, which I run with docker run --privileged -v /path/to/local/files:/root --name cse303dev -it cse303 to mount that local directory.
What is the best way of exporting the container AND all of the contents of that local mounted directory into a simple tar file of some sort? And then how would I easily re-import this container and run it so that I can see and use all of those files copied from the local machine that exported it? Is this possible?
You almost never "export" a container per se. Containers are generally intended to be freely destroyed and recreated.
In your case you already have the data you care about stored outside the container (in a bind-mounted folder, which is the easy case), so you can just copy that directory tree to wherever else and then run a new copy of the container there.
(cd /path/to/local/files; tar cvzf ~/local-files.tar.gz .)
scp local-files.tar.gz there:
ssh there
mkdir files
(cd files; tar xvzf ../local-files.tar.gz)
docker run -v $PWD/files:/root cse303
This is trickier if you're storing the data in a named volume. The Docker documentation describes how to back up the contents of a named volume and you'd have to go through that procedure.