Will Apple TV 4K WiFi work without Cox Communications service?

You can use an Apple TV 4K wi-fi device (or Roku or other streaming devices) for streaming services such as Apple TV+, Netflix, Max, Hulu, etc without having Cox cable TV or any other cable TV service. You would need to continue your Cox Internet service or replace it with some other internet service.

You may find that many “live TV channels” will be exclusive to a pay TV plan, and do not offer a channel-only access nor channel-only subscription through streaming.

If these known TV channels are important to you, then you can opt for a non-cable, non-satellite option: internet streaming TV providers. Largely all of the same TV channel options, but through a different company, and likely at a more competitive price. In the US, you can choose from internet streaming TV providers like Hulu Live, YouTube TV, Sling TV, DirecTV Stream, Philo TV, FuboTV. (Still packages of channels, though.) All of these have apps for the Apple TV box.

Or you can keep the Cox TV plan, and use the Cox Contour TV app for the Apple TV box.

That depends on what each news organization offers via streaming. Two examples are https://www.nbc.com/apps

and https://abc.com/apps but you can do an internet search for other options.