While you are waiting, you may enjoy this conversation between /u/alkibiates415 and myself.

[EDIT: This was the post I was looking for; you should read it too!]

The problem is defining a res publica. That term was used for the state for the totality of the western empire, and a translation was used in Greek for the eastern empire until it was destroyed. I've been thinking about this for a couple years, and while I haven't published, I will say I'm starting to think the term refers less to a form of government than a political community, and I am probably not the first person to think that.

Also, Sulla wasn't dictator for life. That's a popular myth. Sulla's brief was "dictator legibus scribundis et rei publicae constituendae." That is, "Dictator for the writing of laws and establishing the Republic."

While this brief didn't have a set time limit, there's an argument to be made that when the Republic was established, Sulla's brief expired.

In the event, he resigned his commission somewhere between six months and a year into his tenure. I don't know why he did that. If you believe in seances and stuff maybe you can ask him. If I have to guess, I would guess that he was trying to keep to the traditional form of the Dictatorship as closely as he felt possible.

About the "Roman Empire;" Rome had an empire since the Punic 1. By that I mean direct administration of foreign people. That's some 260 years before Augustus.

If you mean "Empire" as in a monarchy, I refer you to the post I linked above.

And again, there's a lot more to say about all of this.