I think that I have at last figured out what the problem was. When creating custom Docker networks, the Docker daemon default address pools would collide with the networking configuration on my Unraid host, and interrupt any network connections I had to the host. I think this is a manifestation of the problem described here: https://noted.lol/fixing-docker-network-problem/ and here: https://old.reddit.com/r/selfhosted/comments/1az6mqa/psa_adjust_your_docker_defaultaddresspool_size/
Those links suggest to configure the default address pools in the /etc/docker/daemon.json but this file is not persistent across reboots on Unraid, so instead I figured out the command-line equivalent, and edited /boot/config/docker.cfg to add DOCKER_OPTS="--default-address-pool base=172.16.0.0/12,size=24"
I've only added this recently at my latest encounter with this issue, so I'm not sure that I'll never encounter this issue again. After trying repeatedly to recreate my Docker image and having multiple unpredictable lockups upon docker network create mynetwork calls, adding this parameter finally let me add all of my custom networks without locking up the server.
I hope that nobody else has to deal with this, but if someone comes across this thread, I hope this solves it for you!