I'm not positive, but I think you may need to say that anyone can enter, even if they don't sign up. But, of course, you can make that fact in very small type !!
If they dont sign up or have subscribed to the terms and privacy statement then any email could be dfined as spam IMO
Firstly are these people that have agreed to your terms and conditions?
Secondly you need to comply with anti spam regulations and offer a way of unsubscribing from within your email
You need to be satisfied you are not breaching the gaming act
http://www.out-law.com/page-6780In addition your privacy policy is insufficient currently to allow you to even market to your customers never mind running a competition.