Finally got our MySQL woes fixed. The issue is that 5.7.14 which is distributed in the Ubuntu 16.04.1 LTS repositories, is broken, at least in our environment with our workload.
I installed MySQL 5.7.15, one point issue newer from the MySQL Community Server repository and it has performed flawlessly thus far.
You can now use MySQL tools on the shell server to connect to and manipulate your databases. The server that has the same version of PHP as the main web server is ubuntu.eskimo.com, which like our web server, is running Ubuntu 16.04.1 LTS.