Dictionary.com
Evidence: that which tends to prove or disprove something; ground for belief; proof.
Proof: evidence sufficient to establish a thing as true, or to produce belief in its truth.
In a discussion, you're usually on your own side of the argument, not your opponents'.
Even still, the non-italicized part points to the simple fact that even going by your line of argument, a body of evidence is a proof.
Yes, there's a one in a jillion chance the next time you sit on it, the chair will in fact break. Or anything non-standard happens. That is an assumption that any self-respecting scientist makes during data analysis and is explicitly included in said analysis numerically in good ones.
The standard is that for not terribly important matters a conclusion that's true in 9995 out of 10000 observations is terribly unlikely to be a result of 9995 mistakes; for DO NOT MESS THAT UP cases like clinical tests it's 9999 out of 10000.
The second part of it is a more philosophical assumption that reality is consistent with itself. It's an entirely groundless assumption, for which there is absolutely zero evidence and zero proof. And yet it's still made - because it can be otherwise summed up as 'knowledge exists'.
It could absolutely be true! Except then we would have to more or less instantly revert to less than cavemen once we realized that. If how the world works, the fundamental laws of physics, can change on a whim then you cannot know anything - what worked as a chair yesterday may be a high explosive today, because someone cranked the cosmic dial on the free energy of wood.
Otherwise, you know fairly reliably that wooden chairs are a decent seating and won't break or vaporize from underneath you if they are the exact same as the chairs you tested and any breaking chairs are breaking due to hidden factors like material damage that you didn't test for but which will also reliably cause the same issue if you do test.
All of the things you claimed cannot be proved have been proposed in the first place by looking at the evidence then proposing a reasonable explanation then looking at more evidence turning up and checking if it, um, checks out. The age of the Earth would have no impact on radiometric dating or whatever else was used to gather evidence on that method; they don't use the age of the Earth as a reference point at all. Even if the age of Earth or Big Bang were
false the evidence that was used for them would still be true; it would just point to another explanation.