That is a myth, the pope never actually died. He simply got buried, then they cremated him under that vatican place, and his smoke came up through the floor into a waiting pope canister. so it si in fact the same pope, and yes, they are already dead when they become popes (look at their facial expressions)
And when the smoke comes out the roof, it is black if the pope is still being decanted and white if he is all burnt into his new form. The wetter a pope is when he dies, the longer it takes for the new pope to be formed.
Also, due to unavoidable loss, popes have shrunk over time - if you looked at the pope 2000 years ago he would be huge! (and yes, they did have popes before Jesus
)