1) How did anyone, including the Greybeards, know Tiber Septim was a dragonborn? There were no dragons around during his time on Tamriel, thus no opportunity to absorb a dragon soul and "unlock" shouts. The only other way to "unlock" a word of power I know of is for one of the Greybeards or Parthurnax to share their understanding of word of power with you. No dragons, no shouting, unless you go study with the Greybeards. Yet the Greybeards summoned him after they realized he was dragonborn. He didn't go there to study and then find out he was dragonborn, but the reverse.

2) How does anybody in Tamriel know that Tiber Septim became a god? There are wayshrines dedicated to him in Cyrodil and Skyrim that confer a blessing, but someone had to build those after he became a god. But how did anyone know he had become a god?

1.) Its incorrect to say that there were no dragons around, because there were. Naffaliarguus in Redguard for instance plus the Dragon you first kill have been around for eons [I know I spelled it wrong but I don't have time to link his name]. As for how did the Greybeards know, well it depends on which chain of events you subscribe too. In some circles (see the Acturian Hersey), all this about Tiber and being Dragonborn (in the sense that the PC is Dragonborn) is a bit debatable.

2.) Because the Imperial goverment said so, and because people believed what they told them.

Except all the dragons that pledged loyalty to him in exchange for their service. Besides, I think the Greybeards called for him way before he even bothered to go to High Hrothgar, and Wulfharth got to him first. If they can hear whispers of words from random word walls, they might be a bit more sensitive when people like Tiber come about. Hes not just a Dragonborn, hes THE Dragonborn.

The man probably made a big scene about it. Or, some priestly orders were born/nurtured during his lifetime, and began to go out and preach his endeavors shortly after his passing. The Blades seemed to be rather receptive to him.