Phone works only if plugged into modem????

You can buy a base station phone system with satellite phones and have your tvs connect to wireless…

another option is reinstall the fiber / gateway to where access is available for your home connections which may not be where you desire the gateway to be placed, example basement, or closet, or far side of home depending on how your residential layout.

A person on site can determine more than asking on a public user forum where none knows the hows or whys involved in your installation.

On the phone issue, is there a wall jack close to the modem?

I did buy a base with satellite phones.  There are phone jacks all over the house as well as near the modem, but none work.  As far as the TV's in the other rooms, I guess I will have to invest in smart TV that is wireless.

So does your TV plug into an Ethernet port? HDMI port? Need to be a bit more specific. If there is a jack by your modem that worked as a phone jack prior to the modem install you can make the rest of the home jacks work only after you've isolated the old incoming drop at the NID from the inside wiring, once you've done that post back. You can google testing from the phone company NID to see how to open the test point than tape it so it can't be plugged back in. Thereby removing the old phone company feed. EDIT: is there crawls space, basement or attic access between the modem and the TV?

 TV, I believe is in one of five Ethernet ports.  I had to buy a splitter so I can connect the phone and answering machine.  I am a math major with a master's degree in applied mathematics.  The previous cables were installed between the walls, but the tech who did the optic fiber install said he could not lower it fromthe attic.  I should have told him to find someone to do the job correctly when he told me I would have to connect the phone into the modem. I need to get their techs to do what you are describing.   

Find a low voltage contractor in your area and have them give you an opinion. Usually, a fire break is only placed where electrical wires are. So I'm assuming your answering machine is not built into your cordless phone system, which means there should be an in and out jack on the answering machine, where you'd go from the modem to the answering machines in or line jack and the phone would plug into the out or phone jack, shouldn't need a splitter follow the installation instructions on your answering machine. Did you do as suggested at the NID?