Usually the date you bought the item or took delivery — whichever your local rules and the warranty terms say. If you bought online and it arrived weeks later, most warranties and consumer laws run from delivery, so keep the delivery confirmation if it matters.

Possibly. A manufacturer warranty is a promise on top of the law, and in many countries consumer-law guarantees require goods to be of acceptable quality and last a reasonable time, which can extend past a short warranty — in New Zealand the Consumer Guarantees Act can cover faulty goods well after the warranty ends. Check your local law and act in writing.

A receipt, order confirmation email, bank or card statement, or the seller's own record. If you have none of these, ask the seller to look up the purchase by your name or email — many will, and a dated record from them still counts.