"Suppose I had a hypothesis that all Earth-era and earlier planets shared some feature that some later planets don't."

Then we can do the reverse approach - crush all the data we do know, and see what changed about the time our Earth came around. We can then take all these candidates, and check whether any seem plausible, then do some further investigation.

More fridge logic:

The Dark Mark. Everything we know about it is that there are very specific restrictions on how the bearer can conceal or display it, and what they can say about it.

Snape, who is subject to the restrictions about what he can say about it, provided information (consistent with all previous information, all of which he is well aware) that Harry, by describing how to use the restrictions on speech to identify bearers, removed those restrictions. Snape then allegedly described the restrictions, and the description he gave is roughly the stupidest possible one that is consistent with all available data.

Which is more likely, that the Dark Lord created a mark that prevents people from saying what it does unless they are told that they will be asked to say what it does, and in that one case ceases all operation... or a Dark Mark that has the restriction "Provide as little true information as possible about the nature of the Dark Mark.", which would force a response identical to the one received.

How does one perform a Bayesian update where A is "The phenomenon produces evidence B in such a way as to minimize the value of P(B|A)/P(B)"?

Has it been pointed out yet that while Hermione lay dying and Harry was trying to save her, he neglected to cover her in the cloak that hides the wearer from death, and also neglected to notice this fact during the time afterwards when he was getting mad at himself for everything he had screwed up?

Suppose I had a hypothesis about something that was present in many Earth-era planets but not astronomically younger planets. How would I falsify such a hypothesis?

Depends on the hypothesis. We have some pretty good picture about the development of the Milky Way. If your theory was about rates of supernovas or a certain proportion of heavy metals, then we could check whether those happened recently. If it was more specifically about Earth-like planets, we could record the hypothesis, and check it in a few years or decades as our picture of Earth-like planets around other stars gets clearer.

Large moons are rare, but very rare? The Pluto/Charon system is even more extreme than the Earth/Moon in that respect, although obviously it formed under different conditions.

Not necessarily. A culture that include the concept of a "raiser" - an octopus with the job of raising the babies, and passing the culture on to them, without mating at all - can avoid that issue. The "raiser" would also improve his average genetic fitness if he is a sibling of one of the parents, since the children would then all have approximately one-quarter of his genes.

(it's unclear why this evolved)

If it's not enough to kill off the species, evolution generally won't drop the feature.

Do you have any candidates for these features? Because there are two ways of seeing your argument: a) There are some very rare features (like the moon) that are necessary for life. This makes intelligent life very rare. b) There are some features that Earth has that are necessary for life, and Earth is one of the earliest planets with these features.

a) explains the Fermi paradox by rarity of intelligence, while b) explains it by saying the first intelligence will expand and prevent other from developing independently. But b) requires that we guess what these features could be.

The one with the power to vanquish the Dark Lord approaches, born to those who have thrice defied him, born as the seventh month dies. And the Dark Lord shall mark him as his equal, But he shall have power the Dark Lord knows not... and either must destroy all but a remnant of the other, for those two different spirits cannot exist in the same world.

Looking at this with my current eyes, I see no reason to anthropomorphize the Dark Lord more than necessary. I think it is reasonable to say that Harry is "born" of the Peverell brothers, who have thrice defied death. Even at this point, death/dementors have marked Harry as an equal (by dealing with him), he has power that they know not, and he has resolved to destroy "all but a remnant" of it.

Yes, that's one explanation. Though that doesn't explain why the Earth is a late Earthlike planet, rather than an early one.

Alternative anthropic principle: There must be a first Type 3 civilization.