If a seller sent an item to a customer via the postal service, the postal service claims to have left it at the house, and the customer claims it has not been received, who is responsible for the loss? for the shipping cost?
From a Halakhic standpoint, there appears to be two questions here:
Regarding #1: If I remember correctly, a buyer only possess an purchased item after he has physically picked it up (kinyan). As such, in this situation, the seller would be responsible for the package. However, see below.
Regarding #2: It appears to me that the postal carrier is most likely a Shomer Sachar, a Paid Guardian. In such case, they are exempt for breakage or accidents, but liable for theft or loss. This package appears to have been lost, and thus, the postal carrier is liable. [Bava Metzia 95]
However, they claim that it was delivered. If one leaves an object in another's yard, the recipient takes possession of it (kinyan chatzer). [Avodah Zarah 71] As such, it would be the buyer's responsibility.
In short, this situation is quite complicated, and would depend on determining exactly what happened, on the contractual obligations of the postal carrier, and on the specifics of the agreement between the buyer and seller (both explicit and implicit) regarding the delivery of the item.
As usual, please ask your Rabbi for practical applications.