Install 2nd modem to existing coax port?

I may get yelled at by ATT for asking this but heres my question.

Is it possible to install a second modem (Arris) in an existing coax outlet in my room? And then hook up a router and either set it up to mirror the original (ATT) modem/router SSID and password. In a sense extending the network?

The original (ATT) modem/router is currently in a very secluded room of the house and provides poor internet throughout the entire house unfortunately. Rather than pay ATT to come move it I am looking to see if:
1. I can myself move the ATT modem to a different coax port in the house and strengthen the connection.
2. Do the method above, set up another modem and router in my room to enable a strong connection in my room.

I do HelpDesk for a living so know mostly what I am doing but I do not know how ATT works. When they activate internet (and u-verse tv service) do they activate all coax ports within the house? If so are they activated only for u-verse tv or only internet or both?

I have a wireless tv receiver in my room and have used the ethernet port on the back but the connection is not ideal for large data transfers on my PC.

Is there a way I can set up my router (which supports AP mode) to act as a range extender either with the wireless receiver or the existing ATT wifi WITHOUT having to run a 50ft cat5 from the downstairs modem?