The 'Telephone Interface Module', with the "In" appears to be the phone/alarm system, NOT ethernet.
RJ31 is generally for alarm systems.
You would go thusly (I think):
Modem -> router -> existing 8 port 10/100 switch (middle image) -> other devices plugged into that switch.
Whoever wired this should have supplied a detailed schematic of exactly what it does. I'm in the middle of planning the wiring of a house, and whoever gets it next will have a full schematic of where everything is.