I think you might need the land line for ADT. Definitely check with them.
NAT router is not a brand. NAT = network address translation
It enables multiple devices to access a single network without wires.
The router requires DSL or cable delivered internet service.
A router is necessary in your house if you want to use even 1 computer without being wired directly into your modem (does DSL use a modem?). The router also has its own firewall, which is an extra layer of protection.
If you want to use wifi on the go, all you have to do is go to a "hotspot", like they have at libraries, coffee shops, and hotels. Or buy a "mobile hotspot":
How to Choose the Fastest, Cheapest, and Most Reliable Mobile Hotspot to Stay Connected Everywhere You Go
The article is a little dated, but I don't think a mobile hotspot is cost effective for routine internet use -- not mine anyway.