Apple quotes that the HDMI port in these can do 4k 240hz

Yes, I have a 4K 144hz gigabyte monitor that runs perfectly with it.

You must get a high quality USB-C cable though. I got Anker brand cable. Previous cable I had lying around the house would only give me 60hz

Depends on the resolution of your external display. I had a 144hz 1080p monitor working as expected with a usb-c to Display Port connection (It didn't work with an HDMI cable).

They do. I am on a last gen M1 Pro MacBook 16" using a 240Hz display but it's capped at 120Hz.