Adding a RJ45 wired extender

(1) Is it possible to connect a RJ45 wired extender to the T-mobile gateway and run to the lower floor. Our ceilings have a old heated mesh (disabled) that prevents the signal to go from floor to floor.

(2) Does the T-mobile gateway port send power thru RJ45?

I would like to add the following UniFi U6 (see link) if it will work.

https://techspecs.ui.com/unifi/wifi/u6-enterprise

Thank you,

Jerry