I don’t think he gets any of them…as one poster, if not more, mentioned - how could he have the time? I DID send him a ‘congratulatory’ upon election - finding that email link was a surprise, so I did it. Don’t recall getting even a form reply back. (I could be wrong.)
As for WRITING the Pope? I once wrote to another Pope in my younger years - pre/during Vatican II changes. I received a Papal Apostolic Blessing (a paper), of which a great fuss was made at home. Then, I believe I received a DUPLICATE Papal blessing. Perhaps, just a form reply. Albeit, they were saved for a long time. But the last time I tried to contact a Pope (more recent and, I believe it was Pope John Paul II) - whatever reply I received appeared to be directed from Washington, DC. Forgive me for saying this or the poor comparison - and please correct me, if wrong - I’m thinking that there’s a special redirection addy (just like letters to Santa Claus end up in post office and not the North Pole) – i.e., mail to the Pope is directed to some Vatican Representative in the U.S.