In some ways I agree with your points Matteo. It could be that the posthumous adoption was just Caesar passing on his name to an heir, nothing more. Though I have read somewhere it's not the usual way of adopting a legal heir. Why posthumously, when most heirs are mostly adopted while alive, if indeed as you say it was as simple as a Roman aristocrat wanting a 'legal son' to carry his name. Hence me thinking it was something to do with Octavian's safety (and Caesar's too, I mention the example of Augustus having to deny an heir in his will after recovering from illness). Still, you could be correct too, in that Caesar just considered it a family matter and soemthing he wanted to pass to Octavian in case of anything, after all he was on the way to another campaign in Parthia.
However, Caesar as "more republican than Octavian"....well he certainly didn't show it in the run up to his assasination. He styled himself in the garbs of ancient kings, failed to show proper respect to the Senate and at least one source (Cicero if I recall) suggested that the whole thing with the crown and Mark Antony was planned to gauge how much the crowd would ahve supported him being king (as it was, not much support!).
Octavian, even after becoming Augustus, showed much more respect for traditions of teh republic, especially the Senate. As you said, not even Octavin imagined he would become "emperor", likely at the beginning what Octavian did was simply to insist on what he thought was legally his, the name of Caesar and recognition as legal heir of Julius Caesar. It just so happens that demanding those rights led him down the path where he had to put down enemies one by one, including those who would consider themselves equally Caesar's heirs politically/militarily.
As for the son, I think the more interesting one would be for Calpurnia to conceive after Caesar returns from Gaul/crossed the Rubicon, and the son is born while Caesar was fighting Pompey etc. I don't think it'd butterfly away Caesarion, Caesar could still enjoy Cleo as mistress methinks. But after that, he wouldn't have the posthumous adoption of Octavius, no? Similarly, his focus on Rome now is celebrating that he has a son, a ROMAN son! No one would giver a damn anymore about his son with Cleopatra, people would just nudge and wink and make jokes, but that's about it, no?
So...would he still want to go to Parthia, just like OTL if he wasn't killed? Would the conspiracy still happen, "king Caesar" now has an heir after all. Caesar may still support Octavius' career, just like he did for many of his relatives (I think Mark Antony is like a distant cousin of Caesar's right?), might even help him reach all the way to consulship. If Caesar still died with his son as infant, would Calpurnia remarry? And with who, and could that person be an important "kingmaker" among the Romans, heh heh.
BTW, would the son also be named Gaius Julius Caesar, or is there some other possible first name (Lucius? Quintus? Marcus?).