I assume you have a LAN wiring closet defined, where all rooms pull their cables to a common location.
This is the location for the top level switch. Power is required here.
The modem and router can be located together at the cable MPOE which can be elsewhere.
A single drop from the MPOE location to the wiring closet is required for the router private side connection.
A switch in every room will not violate the Ethernet 3-2-1 rule. No worries there.
I doubt the house is large enough to violate the 185 meter end-to-end length rule, either.
Traffic between rooms will switch across the backplane of the top level switch (in the closet).
Traffic bound for the Internet will hop to the router, modem and beyond.
With CAT6 wiring, you can run GB Ethernet at full duplex.
Are you pulling phone cables to the rooms also?