questions about connecting Apple TV 2022 via Ethernet cable

The Apple TV 4K (2022) comes in two flavours - the 64GB (Wi-Fi model) and 128GB (Wi-Fi + ethernet model). See the Apple TV 4K - Technical Specifications. Clearly you can't connect the first one with ethernet, so I presume you have the second model.

As you can see in Apple's instructions Set up your Apple TV all that you need to do is:

• Connect your Apple TV 4k to your router (Wi-Fi or ethernet)
• Connect your television to your Apple TV 4K with an HDMI cable

Observe Apple's note about HDMI: On Apple TV 4K, you need an HDMI 2.0 or later compatible cable. To access all of the features that your Apple TV has to offer, use an HDMI 2.1 cable.

To use it with the LG TV rather than wireless, do I hook an Ethernet cable from the Apple TV to the router? And then run an Ethernet cable from the Router to the TV?

Apple TV will need its own network connection (ethernet or Wi-Fi), and the TV often also wants its own network connection (ethernet or Wi-Fi) when used without Apple TV. If the router is some distance away, then you can use a single ethernet cable for the long distance, then use an (unmanaged) network switch for the last bit.