So, to start off plainly and bluntly, the show takes a great deal of liberties across the board, and furthermore it is worth mentioning that the sourcing they do do isn't necessarily academically sound sourcing - that is to say they may be taking sources at face value that historians would see as heavily biased, suspect or shaky, or taking something that might be said as "this is how it might have been" and reading it as "this is how it was." It is a historical drama first and foremost, and as the creator himself stated it aimed to "feel authentic" rather than "be accurate" and was designed around popular expectations and balancing drama and plot over the, well, sometimes dry nature of reality and history. You're watching a program designed to entertain rather than to inform at the end of the day. By all means it's a good show and is certainly entertaining, but don't take its depictions as accurate or even its plot as historical fact. Characters and motivations are completely fabricated at will, most notably Lucius Vorenus and Titus Pullo are essentially mentioned by Caesar in passing, all of their adventures and involvement is completely, utterly fictional. Likewise, characterizations of the actual historical figures tend to differ from the way sources have described them.

So with that big cat out of the bag I'll get to the three specific points you were curious of. As mentioned, the exploits of the leads are entirely made up, even more so here we have very little to no record of Octavian's early years, so not only is his kidnapping made up, but also his entire trip to Gaul. Likewise, there is no record of the whole aquila plotline, it was presumably made up due to the famous lengths some generals went to recover aquilae and the semi-religious nature of legionary iconography making good plot motivation. As I'm sure everything so far has foreshadowed, no, that characterization of Octavian isn't historically supported either. As mentioned, there isn't really much as far as records to go off of for his childhood, so with that means no real characterization of a young Octavian. As for adult Octavian, he was described as a populist and, by Suetonius, fond of other men's wives. Sir Ronald Syme characterized him as a "chill terrorist," outwardly calm, collected but a quite prolific schemer. Now, it is important to emphasize that I am not stating that young Octavian was not a spoiled or rude child, rather that we have very little idea what Octavian was like as a child, other than that he quite impressed Julius Caesar in his trip to and time on the front with him, culminating in him appointing Octavian as his heir.

So, to recap, it's 0 for 3, all of these things were made up, one way or another. Since we're talking about Octavian another interesting divergence is in his name itself. Octavian would have never referred to himself as, or stood for being referred to as, Octavian. It revealed his plebeian heritage, part of our usage of the name originates from Cicero constantly referring to him as Octavian, a point of belittlement/amusement. He used the cognomen Thurinus prior to his adoption, and then following naming traditions he took on Julius Caesar's name. When differentiating them, he preferred to use the addition "Divi Filius," following his establishment of the Temple of Caesar. Though some, like Cicero colloquially used Octavianus (anglicized now to Octavian) to highlight his plebian origin. Later, of course he was granted the additional name of Augustus. In reality, anyone referring to him as Octavian would be doing so as an insult or jab, but for familiarity's sake the show does not really use Roman naming customs