When I did a lot of hand surgery, patients did indeed complain of greater pain from old injuries during the cold winter. I have not heard that from my rhinoplasty patients, however. Anything can occur after a surgery, but I would think this risk is relatively low versus the benefits of an indicated rhinoplasty.
No, this is not true. If this was a problem I would never recommend a rhinoplasty. There are millions of people who do not live in the tropics after this procedure.