Send this request in the final week of your notice period rather than after your last day. Once you are off the payroll and your email is deactivated, the request has to route through someone else, and it slows everything down.
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.