Can I use T-mobile Home with or instead of my Airport Extreme to have a network backup system?

if you mean for timemachine then I don't think unless it's listed that it would support it, apart from that then the extreme does not really offer any other unique features

your options are

1. connect the ext hd to the mac you wish to backup using USB
2. connect the extreme to the tmobile home and keep using the extreme, you need not replace anything really
3. look into a NAS that support timemachine I believe my synology nas's support that

unless the tmobile device has support for time machine using that alone is not an option if you wish to use wireless timemachine

you can just connect the airport express to the carrier router and not use the carrier user as anything but a gateway and continue to use the airport express as you do now

or as stated get a nas that is connected to the tmobile router which support timemachine backup

T-Mobil's web site did not provide much in the way of details on their gateway.

The big questions.......

1) Does the T-Mobile device have a USB port to connect your hard drive?

2) Does the T-Mobile device support Time Machine?

Very few routers and gateways support Time Machine, even if they have a USB port to connect a hard drive. I would be surprised if the T-Mobile device supports this type of backup setup.

A "nas" is a Network-Attached Storage device. It is basically a stand-alone computer with built-in HDDs. An example would be the Synology NAS line: https://www.synology.com/en-us/support/nas_selector