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.
Signatures if You're Emailing. If you are sending your cover letter or inquiry letter by email, end with a polite sign_off followed by your full name. You do not need to sign a cover letter that is being sent electronically. Write out your full name in the same font as the rest of the letter, and there's no need for italics or a handwriting font. The formatting here is very similar to an uploaded cover letter. However, emails do not contain a header with your phone number or other contact information. It's a good idea to include these details in your closing paragraph or after your typed signature. That makes it easy for the employer or networking contact to get in touch with you.