You assume Aegon is fake but there's really no reason to believe that, given the story. He's got the purple eyes. Substitute kid murder has already been established as something that happens from time to time. Unless Jon Connington has somehow managed to wrangle a Faceless One's services (for non-assassination purposes?), and let's face it - he doesn't have it "like that", then Aegon has to be real. Sure, it seems out of nowhere, and their efforts are certainly doomed to fail in hilarious ways, but that seems to make it more valid to me - that Connington would only make these real desperate moves if he really believed in Aegon. And why wouldn't he?
Carrie is absolutely correct that Valyrian features are far, far, far from uncommon throughout the former colonies of the Valyrian Freehold. Lys in particular is noted for possessing strongly Valyrian features. The question of Aegon's lineage is that he's a member of
House Targaryen. Specifically, that he's a specific member of House Targaryen who is the son of Rhaegar and a claimant to the throne.
But lets look at his companions. Jon Connington believes in Aegon, sure, but what reason does he have to believe that other than his feeling that he failed Rhaegar and wants to do better for his son? Really that could just as easily suggest that he's been misinformed that Aegon is real entirely because it's powerful motivation to protect him and promote Varys' interests.
But look at the other followers. The Golden Company have been the home of bastard pretenders to the throne of Westeros since their inception. It's populated by dispossessed houses that are introduced as people trying to return home. Their leader is even known as "Homeless Harry", to hammer the point in. These people are strongly influenced to return to specifically Westeros, and it's extremely convenient that they're engaged in the Targaryen Reconquista under Aegon when that's what they already wanted to do.
But really, this plays into the opinion Varys expresses that power resides where men think it does. It doesn't, fundamentally, matter if Aegon is the trueborn heir of Rhaegar, or a likely babe stolen from Pisswater Bend. He's been raised as a King and he acts like a King, and that's both very Varys and also a strong indication of why Aegon is likely to be faked. It's also exceptionally convenient that Aegon has a better claim to the throne than Dany, being a male child of Aerys' eldest child. This all plays right into Martin's modus operandi of telling a story of how feudalism is ridiculous.
Of course, Carrie is also entirely correct that the complete omission of the Aegon plot in the show suggests that Aegon isn't real and the plot goes nowhere, except to maybe be a million times more plausible way for Dorne and Highgarden to arrange loyalty to House Targaryen again without travelling to Slaver's Bay.
There's nothing to suggest Dany doesn't know at least the basic truths about her past. She lived in the house with the Red Door and was told all about her family from her brother and the guy that raised her for some time. Hell, she remembers the names of all the dragons her brother kept talking about; there's probably a lot more she knows that the story hasn't filled us in on
There absolutely is. The "Usurper's Hired Knives" are a massive glaring example of where Dany's knowledge of events does not match reality at all. We know from book one that Jon Arryn strongly counselled Robert against attempting to kill Viserys and Dany and Robert bitterly complains that they'd gone out of his reach before he even tried sending someone to kill them. Dany's only encounters with assassins are after this discussion, when the choice is taken to try to have Dany killed based on Jorah's information. However, Dany repeatedly talks about or remembers running from "the Usurper's Hired Knives". Now either they're running from other assassins, or there were no assassins and Viserys was lying to her. One or the other. Either way, her recollection of her past is incomplete. Her memory of a lemon tree in Braavos is categorically infeasible, and the revelation of Arianne's secret betrothal to Viserys suggests a much stronger likelihood she was in Dorne at the time, possibly around the time Oberyn tried to raise Dorne for...Viserys. Interestingly, Jon Arryn was the guy who defused that problem as well.
Sadly, a lot of the pov characters for these events are now dead. It's a device used to keep us from categorical information about the situation, but it does suggest that Dany's early life was not spent running from Robert's men, but for some other reason.