Gonna have to keep this brief but I hope I can answer all your questions.

Was the Eastern Roman Emperor (Byzantine) a de jure monarch?

The concept “de jure“ is difficult to apply to ancient societies. De facto, the Roman Empire had been a monarchy since Augustus and everyone was aware of this as early as Tiberius‘ accession at least. While the Emperor still had to maintain the consensum universorum by balancing the needs of Senate, army and plebs urbana, it was openly known that the Imperium Romanum had become autocratically ruled. Everyone preferred this to the decades of warlord rule and civil war in the Late Republic. However, the position of princeps was not a legal “office“ for a long time.

Justinian made attempts to legally codify the position of the Emperor in the Eastern Empire in the 6th century in which it was made very clear that the Emperor was autocrat and sole ruler, but he was pretty much just putting into writing what was already praxis for centuries. So yes, the Byzantine Emperors were de jure and de facto monarchs and no one questioned this.

how does the title Basileus, which was used by the the eastern Roman emperor as a title, which I believe translates into English as "king", as it was also used by Alexander of Macedon relate to this theoretical question?

Basileus was in fact an ancient Greek word for king, but it was also used for the Roman Emperors in the Greek-speaking East since antiquity. Heracleius was the first Emperor to officially adopt this title in 629 instead of σεβαστός (Augustus). When the Byzantines used the word “Basileus“ we know they meant “Emperor“.

Furthermore, was say Alexios Komnenos as an example as Roman as Julius Caesar, Octavius Caesar, or Sulla?

This question is somewhat confusing. What does it mean to be “more Roman“? I guess this depends on what you’d call integral parts of being Roman. Alexios Komnenos was a Roman Emperor ruling over the Roman people from Constantinople, one of the Roman capitals since Constantinus and the sole capital of the Roman Empire since 476. He was one Roman Emperor in an unbroken line of Emperors since Augustus (and in the East, since Arcadius 395). Sure, the conception of what was Roman had changed massively since antiquity, but just because he didn‘t wear a toga (mostly went out of fashion as early as the 3rd century) or spoke Greek (Greek was always the dominant language in the hellenized East even during the Principate) or was Christian (many Romans had been since the first century A.D., all Roman Emperors save for one had been since 324 and the entire Empire officially was Christian since 380) doesn‘t not make him Roman.

Hope this helps!

Sources:

Egon Flaig (1992): Den Kaiser Herausfordern. Usurpationen im römischen Reich.

Anthony Kaldellis (2019): Romanland: Ethnicity and Empire in Byzantium.

Cyril Mango (2002): Oxford History of Byzantium.

Rene Pfeilschifter (2014): Die Spätantike. Der eine Gott und die vielen Herrscher.

Rene Pfeilschifter (2013): Der Kaiser und Konstantinopel. Kommunikation und Konfliktaustrag in einer spätantiken Metropole.