The ethernet connects from the ONT and is you WAN connection to the internet. If your internet speed is 150 mbps or above the connection to the router has to be on ethernet.
The coax connects to the router to create a MoCA LAN Network for your STBs and FNE. If the coax light is not on your FNE that means it is not getting a signal from the coax. Make sure all the coax in your house is connected at the splitter.
The Fios TV One main doesn’t necessarily need to be next the G1100 but it does need to be connected to the coax. Only the minis can be wireless.