Does the new Apple TV (4th gen) HDMI port ethernet enabled? Should I purchase high-speed with ethernet HDMI cable?

The HDMI port connects to your TV monitor.
The ethernet port can connect to your internet router via ethernet cable. Some cable boxes, like the ATT U-Verse box, have an ethernet cable you can plug into as well and connect to the internet that way if you subscribe to their data plan.

If you have ethernet connected, then your aTV4 won't be connected via WiFi, but depending on the topology of your home's network, that might not matter whatsoever. For instance, I have my aTV4 plugged into my ATT U-Verse box, but I also have an Apple WiFi system for my home. I can still connect via WiFi from say, my iPhone or iPad to the WiFi network and find my aTV4 to do things like streaming via Airplay.

The 4th gen Apple TV HDMI is version 1.4 which technically should support HEC (HDMI Ethernet Channel) as long as the TV or Receiver supports it and you are using the correct cables. But I make no promises as I have no way of confirming this.