re: Was Caesar Augustus gay?

Yeah, I think he was probably first referred to as Gaius Octavius, or less formally as Octavius.

Then he was probably referred to as Caesar in memory of his adoptive father and to preserve his line in the minds of legions he commanded.

Then he was referred to as Imperator by his army, though probably still Caesar by most others.

Finally, he was probably referred to as Augustus, probably referring to the connotation of that word as "Sacred One" or something similar. Perhaps "Majestic."

I doubt he went by Princeps after defeating Marc Antony and Cleopatra. No one was under any delusions at that point. Of course, this is all speculation on my part.