[mothered]

I think that the company that is engaged in receiving parcels and sending them to Russia is very worried about its reputation and asks couriers to take photos or ask them to sign some kind of document. 

Photographic evidence can easily be circumvented.

In terms of signing a document, Is that between yourself and the driver on receipt of goods? If so, what does the document Involve?

+ Amazon knows that the parcel went to such a company. 

It has nothing to do with how you accept your package.