I didn't see it mentioned, so let me hit on something here. When you apply to a law enforcement agency, part of the background process is having you sign a privacy waiver that allows the agency to look into any part of your life. This includes Facebook. Most people don't know that Facebook has a section dedicated solely to matters of this nature. So if you have a "private" Facebook, don't assume your investigating agency won't be able to get into it. They simply need to fax over your signed privacy waiver and voila, your account is unlocked to them. When you think about it, as big as Facebook is, it's not surprising at all that they would have a section dedicated to this. My suggestion is this: Surround yourself with decent people. Tell your friends not to put up pictures of you (You can change this in your settings, so that you have to allow them to be able to "tag" you in a photo). If you've got anyone on your Facebook that you wouldn't want your investigating agency to know about, GET RID OF THEM. They're not worth having as friends.
If your investigating agency absolutely prohibits Facebook, than it should be deleted. Don't think you can bypass the requirement by using a fake name and/or email address. It WILL come back to bite you in the rear. Think ahead to the future when you're taking your polygraph or voice stress analysis test - there's one magic question on there that is a catch-all: "Were you honest and truthful with all of your answers on your application and during the background investigation?".
Good luck, and like I said, the best way to ensure that Facebook doesn't sink you, is to either delete it altogether, or make sure your "friends" and your photos don't show you in a negative light.