I don't agree with the social democracy thing - it is basically a movement, a political party.
Then, if they're winning in most of the post-communist states, that's another matter. But that countries have democratic republics now.
Putting a "social democratic republic" in would mean adding "federal republic" and many other insignificant varations.
On the empire, at first I tought of adding it, but it is basically a different name for large monarchies. Apart from the fact that an empire is spreaded through large territories and have a lot of conquered territories, there is no difference with monarchy.
Just think: the Queen of England was at the same time the ruler of the British Empire.
About the roman empire thing, it could be both a despotism or a theocracy. After all, the Roman emperor was a "pontefix maximus" (I don't know the english word, please tell me) at the same time.