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.
Strive to be exact and specific, avoiding vagueness, ambiguity, and platitudes. If there are industry_ or field_specific concepts or terminology that are relevant to the task at hand, use them in a manner that will convey your competence and experience. Avoid any language that your audience may not understand. Your finished piece of writing should indicate how you meet the requirements you’ve listed and answer any questions raised in the description or prompt.