Actually it was called "Rhomania," and they continued to identify as a republic/res publica/politeia rather than switch to calling themselves an empire after the reign of Augustus.
Ῥωμανία can be transliterated as "Romania" or "Rhomania". "Rhomania" is less common.
Arguing about whether the Roman Empire called itself an 'empire' is kind of nonsensical. They called it an imperium, or a basileia, fairly regularly, but neither of those terms was intrinsically non-republican; various republican officials held imperium as well. I don't think that any academic seriously contests that the Empire from Augustus onward was a military dictatorship and a monarchy, and that this was a meaningful change that fundamentally altered the levers of power in the state. Sure, it retained some of the institutions of a Republic, in altered forms, but so did men like Jiang Jieshi. Republicanism was a talisman for a long time with meaningful symbology, true, but so what? The existence of Saddam Husayn's Republican Guard didn't make Iraq anything other than a dictatorship.
Later Romans were actually fairly democratic, as their emperors would not often dare execute major initiatives without first announcing the idea in the hippodrome and receiving applause. If the crowd instead voiced its displeasure, the emperor would tend to present it as someone else's idea that he too opposed. Once power had retreated behind the walls of Constantinople, the emperors had much more to fear from the locals rioting than from any external enemies.
There were actually quite a few bloodless coups. Rebellions were almost just elections, with armed rebel leaders parading towards the palace just hoping to inspire the mob to side with them and demonstrate that the will of the people and thus of God as to who should rule them had changed.
Not...really. Acclamation, when it was sought, could usually be orchestrated in advance. To go back to the Saddam example, it was worth about as much as those 99% favorable ballots. Or, to use a more contemporary example, it's like how most Hellenistic monarchs, when going to "allied" poleis, would be "voted" various honors by the
bouleutai, when everybody knew that the guy with the army gets the honors whether they wanted him to get them or not. There's no meaningful element of democracy there.
Coups were only bloodless when the victim was unwilling to fight them, or unable; while the mood of the Constantinopolitan populace might be turned in favor of one side or another, that had basically nothing to do with an election. You might as well say that tsarist Russia was democratic because Nikolai II abdicated without a fight in 1917. And there were an awful lot of coups that were not bloodless, and certainly not decided by the people of Constantinople, or any other city for that matter. What was democratic about the civil wars of the 980s, or the so-called "Seven Revolutions" at the dawn of the eighth century?
That was a fact I appreciated most about a book I read a week or so ago, The War of the Three Gods: Rome, Persia, and the Rise of Islam. I don't think the author ever once used the word Byzantine, though the Rome in question was the 'new' one. Brownsworth is only using it because he's writing to completely oblivious audiences who think "Rome" fell in the mid-fifth century. I purchased it because Brownsworth's podcast re-ignited my interest in eastern Rome a couple of years ago, and it seemed like paying a due.
This is a fairly obnoxious trope, of which, admittedly, I've been guilty in the past. Academics in English rarely use the word "Roman" to describe the Byzantine Empire
unless it's to emphasize some aspect of continuity with the thing that most people recognize as the Roman Empire. Yes, there were some aspects of political continuity between the OG Roman Empire and the Byzantine one. There were also some very severe
discontinuities between them. And it inhibits general understanding when the vast majority of English-language discussion on the Byzantine Empire talks about it as the freakin' Byzantine Empire, not the Roman Empire.
So yes, it is true that the people of the Byzantine Empire mostly called themselves Romans and said that they lived in the Roman Empire. They sometimes also called themselves Greeks, and people outside the Empire at the time would refer to them as Greeks, Romans, and, yes, Byzantines. But march into a Byzantinist's office and start talking about the Romans, and she'll probably roll her eyes.
Which is kind of the point. Brownworth wasn't an academic when he made his podcast or wrote his book, he was a high-school teacher.