A relieving letter request email should confirm three things in the opening lines: the date you resigned, your last working day, and that your handover is complete. Those are exactly the facts HR checks before releasing the document.
In your last week, not after you leave. Once your corporate email is deactivated your request has to go through a personal address, which usually adds delay.
Ask for whatever they will issue in writing — an acceptance-of-resignation email, an experience letter, or a full and final settlement statement. Most new employers accept a combination when a formal relieving letter is unavailable.
Only with your employer's agreement. Many contracts allow you to pay the balance of the notice period in lieu — offering that up front is usually what unlocks an early release.