When you start the server with -nogui you can still enter commands (if the terminal is still attached). Just type the command and hit enter.
You can start a debug profiling session from the console or using RCON.
If you are able to, running the built-in debug profiler and/or a dedicated profiler (like VisualVM) and attaching the results would also be very useful in order to find the cause.
Yes, same for the hosting person.
You should be using Java 8.
Not applicable in latest 1.16 snapshot. Furnace data format appears to have been changed.