They move onto private land!!! At least in Colorado the do
You are correct on all three. Elk have no problem moving long distances during the night. So they will move down to deep dark nasty places where the ground in moist and cool. Or they will move up high where the air is cooler. You will find them more active in the am as it stays cool longer (during daylight hours). Then again just before the sun drops down. During the day, when it's hot, they will be in dark, nasty thick places that are hard to walk in without the whole county knowing your there.
Oh, and if it's windy. Sneek into places that are protected from the wind. Little hidden places where they can still hear you coming a mile away.
This is why elk hunting is so fun!!! It's amazing how an animal so large can hide so well.