That moon-sun apparent size thing is weird; astronomers haven't discovered another planet with moons like it. (Then again, that isn't too surprising – planets are hard enough to find without finding moons too – but it slightly reinforces how bizarre this whole thing is.)

I chalk it down to coincidence, but that's just 'cause I'm boring, and can't think of any more plausible hypotheses. But it's one of the things to watch; it's entirely possible that we discover something in the future that raises a hypothesis to be more probable than Occam's Razor suggests for the "coincidence" hypothesis.

Hidden history? Well, I'd probably make a bet with odds of 5:1 or above: I think there's about ⅙ of a chance of some isolated civilisation, of a similar advancement to our last century's civilisation in some respect (probably philosophy or maths or literature or something else theoretical), having existed. But that chance is only dropping with time, as we get more advanced and so raise the bar (90 years ago we were more "advanced" than 100 years ago, etc.), and time goes on and we leave fewer places to be searched.

I doubt very much that there's a conspiracy to hide some big globe-spanning advanced technological civilisation of the past from us, because one existing in the first place and then having disappeared without wiping out all humans is unlikely (although, Toba catastrophe theory makes this one around 5%, or 1 in 20), let alone while leaving sufficiently small amounts of a trace for its existence to not already be known to us (a hypothetical technological society accidentally self-destructing – or even deliberately, if it was anything like ours – would leave traces everywhere; even if all humans deliberately deconstructed our places of work, followed by our infrastructure, followed by our own homes, followed by suicide, we would still leave loads of traces like plastics and having used up the fossil fuels and circuit boards and stuff); assuming that the entire civilisation was green and eco-friendly and produced almost entirely bio-degradable stuff to the extent that there's almost no evidence of their existence… Given that it's humans running the scene, and that I've only got experience from the one civilisation I know about to go by, I'd generously rate it about 0.1% likely that a hypothetical civilisation would take this route (one in a thousand hypothetical civilisations would do this, I reckon). And then there would have to be evidence found by a select group of people, which, given how hard it is to find such evidence buuutttt the near-certainty that some artefacts would still exist but that we've hardly got anyone looking… 4% (1 in 25). And then that they stayed reasonably quiet about that discovery (an optimistic 5%, but only because it's conceivable that the sort of livelihood / hobby that would permit this discovery selects for people that would not write in to tabloids) but they published it somewhere (in a small scientific paper, on an obscure forum somewhere if the discovery was recent, etc.; 95%, I reckon), that conspirators (their existence a very generous 40%, because if they existed I wouldn't know) who cared about this issue (90%, because I can see why some hypothetical conspiracies wouldn't want the political instability but I can see why many conspiracies just wouldn't care at all, but I'm taking into account that there might be multiple conspiracies) would find it (given that it has to be obscure, but that the conspiracy would be monitoring known sources, but that this wouldn't be the only thing they cared about and they'd be assigning it a reasonably remote possibility because it's pretty unlikely on the face of it, but I'm giving them an insanity bonus and assuming multiple conspiracies, 10% chance of finding this… actually, scratch that; I'll give them 40% chance, because there can't be that many sources to trawl through, unless this discovery was reported in the Internet age in which case they could write a program to search for them), shut it down (this is conspiracies we're talking about; who knows how much power / influence / convincing ability they have? 80% chance of success), set up a worldwide network to monitor for new discoveries (95% chance, because we've already assumed they've got sufficient resources) and shut them down (90% success chance for shutting all of these down, because they're clearly putting more resources into this and catching it earlier).

Multiplying all of these made-up-but-reasonable-sounding probabilities together to find the total estimated chance of this scenario occurring? 5% × 0.1% × 4% × 5% × 95% × 40% × 90% × 40% × 80% × 95% × 90% = 0.0000009357%. Even assuming that I've overlooked a thousand possible ways that this could happen, this is not even worth considering the possibility of given the evidence – unless you know something you're not telling me.