Irrelevancy and the non-sequitur fallacy

You have a few different questions that have different answers.

We know that simply wanting something to be true does not make it so. But…  how exactly do we know this? 

Regarding fallacies, it doesn't matter. If the reason given is "because I want it to be true" it is fallacious. If I say that the earth is NOT flat because I really want it to be not flat... this is fallacious. It doesn't matter if the earth is flat or not.

How do we know that plausibility does not imply correctness? Or how do we even know that correlation does not imply causation? 

We don't need to know it doesn't; we just need evidence that it does . This is the idea of burden of proof . For example, if one claims that vaccines cause autism because shortly after giving their child the MMR vaccine(s) the child was diagnosed on the spectrum, in their mind they have established plausibility and objectively, correlation has been established. We cannot claim causation until it has been properly established (in science, establishing causation requires strict and specific methodology). We can never really say with certainty that there is no causation but we can say that there is no causation based on probability (e.g., we are extremely confident that vaccines don't cause autism based on exhaustive research combined with the fact that there is no plausible mechanism ).

In summary, it is fallacious to claim causation when it has not been properly established.

We know that simply wanting something to be true does not make it so. But…  how exactly do we know this?

Test it out.

Wish for $1 million, in cash, to come raining through the roof of your house.

It won't happen, because wishing for something to be true (for money to fall from the sky into your hands) doesn't alter the reality (that money won't fall from the sky into your hands). There's no causal relationship between wishing for X and X happening, because 'wishes' by themselves have no causal power.

How do we know that plausibility does not imply correctness?

There are many possibilities, fewer probabilities, and a handful of certainties. Some of these possibilities are mutually exclusive. If plausibility (or possibility) implied correctness, then you'd have to grapple with blatantly contradictory outcomes.

For instance - I am thinking of one number between 0 and 10.

It is possible that this number is 2. It is also possible that it's 7. Yet I am only thinking of one number, so it can't be both 2 and 7. Not to mention the fact that there are numbers other than 2 and 7, between 0 and 10, that I could be thinking of.

Or how do we even know that correlation does not imply causation? How do we determine any of this?

Lots of things correlate, but there's no evidence that they are causing each other, nor is there a plausible mechanism for them to be causally related.