What do you find at the front of an elephant and the back of a horse?

the letter "E"

What is as big as a horse but weighs nothing at all?

a horse's shadow

What is a foal after it's 4 days old?

five days old

How far can a horse run into the woods?

halfway, then it's running out of the woods

What side of the horse has the most hair?

the outside

A cowboy rode into town on January 5th. He stayed for a week and then rode out again on January 5th, how is this possible?