The competition is not running yet - it soon will be.
No, they don't need to Like any page, they need to create something and send a picture of it to webmaster@...... by any email method they want. As long as I receive their email with a picture then they will be entered into the competition.
I'll be interested to know if anyone can come up with a solution to extracting a list of all FB likes.
But if you just want to select a random winner from one of those who have liked your page then simply check how many likes you have within a date range (easy on FB) and then select a random number. Match that random number to your list of likes and you have a winner.
You can notify the winner by leaving a message on their FB page.
Why do you want to email all the FB 'like' people? Simply say in the competition that the winner will announced on your FB page on a specific date - that will invcrease the number of FB views.