Return address and date: (Upper right hand corner) Once upon a time, this was necessary in case the letter became separated from the envelope, but I doubt that happens much any more. Even so, this is where your reader is going to look to find your address and the date the letter was written, so it should be there. Mailing address of recipient: (Left_hand side, one line below the return address) This identifies the recipient of the letter. In office environments, letters are often removed from their envelopes and circulated; this helps the letter get back to its rightful recipient.
How do you know for sure whether or not to use a closing? One good test is to ask yourself whether this email is more akin to an instant message/text or a business letter. If you're giving your teammate a quick update on an ongoing project, a formal closing might not be necessary; if you're throwing your hat in the ring for a promotion, it definitely is required. When all else fails, and you're still not sure, err on the side of caution and include it. You'll never go wrong by being too polite and respectful.