Does the head dong thing get sunburned? Does it convey a lot of heat?

The point of a hat is to 1) keep sun out of eyes, and 2) keep the head cool by blocking the thermal radiation that is sunlight.

So, picture 2 fails at both of those. Picture 1 is fine if the head dong thing is just unsunburnable horn and self-regulates its own temperature, plus it’s less prone to being blown off by the wind.

If the head dong thing needs to keep cool then picture 3 is the way to go.

No, you tell the time by staring into the parts of the sky that aren’t the sun, and use that to figure out where the sun is.