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.
Letter Paragraphs: Professional letter paragraphs should be relatively concise. The first paragraph may include a brief friendly opening and a concise explanation of your reason for writing. The second paragraph (and any subsequent paragraphs) should expand upon your reason for writing. The final paragraph should restate your reason for writing and, if applicable, state your plan of action (or request some type of action be taken by the reader).