What are the challenges of empire and what is the usual result?

It depends. if we're talking about the emperor it self, than i dont know the direct answer. If we're talking about a large scale empire, like the Roman Empire, than first of all, there are multiple religions, cultural groups, a huge area of land to keep safe and guarded, and you have to make sure that your people like your empire, so they wont go nuts and start revolting all around your country. For the roman empire, barbarians were a problem too. The usual result is either 1 of 2.

in the first case, the emperor can govern the country just right and make sure his people wont go nuts for a while, take care of foreign business well, to make sure he's friends with neighbours and have a strong currency, and army, to keep the economy and the country safe. This usually works for sometime, but after a few or lot of years the people will go nuts, because things always change.

in the second case, the empire falls apart slowly and basaically get eaten by other countries who surround them or who got independent after a revolt.

with this im basically saying, there are a lot of problems with medieval or earlier empires, and no matter what the emperor/empress does, it will fall apart somehow.

Hope I gave a somewhat good answer