> Are entries drawn randomly from all entries received until all slots are filled?

Yes. All entries are drawn randomly. If date of first choice is filled, it tries your second choice, down the line. If one of your choices is successful, any other entries you might have entered are then marked unsuccessful.

> Do you have an advantage if your first choice is a date with the least first choice entries?

No, no advantage like that.

You are sure to be successful if you pick a May or October date, but then the hike might not be successful.

The one way to get an advantage is to make multiple applications at $6 each. You can enter as many times as you want, it just requires more time and more money.