I have a setup similar to this but in Australia the internet comes into the house via HFC (coax). Given that there are Coax outlets in every room, am I right in thinking that I plug in the router in a given room, and then all ethernet points will work?

Can you also get a switch for coax cables too? In Australia the internet comes into houses via coax cable. the house we’re moving into has Ethernet outlets and coax outlets in each room.