Applying for a posted internal role means competing with external candidates on paper. Use the letter to do what they cannot: name specific company projects, systems and relationships you already own.
Both. Raise it in a one-to-one first, then send the letter as the written record your manager can forward to HR and their own manager.
Measurable achievements, responsibilities you have taken on beyond your job description, and a direct comparison to the target role's requirements. Tenure alone is the weakest possible argument.
After a strong performance review, at the close of a successful project, or during the company's planning cycle when headcount and budget are actually being decided.