1. No, you do not need a receipt from post office. It will not help you in any case. So you can simply drop-off your packages at the post office.
2. Postage you print through amazon will not cover you any more that postage paid at the counter. You just saving some time by doing that. To be protected from "undelivered" claim you need "signature required" service.
3. your buyer confused and is probably member of Prime. So he gets free 2 day shipping.
But it only applies to Amazon fulfilled items.
Unfortunately not much you can do here except trying to politely explain it to customer. If you get a negative try to remove it.