Mathematical proofs are in escapable conclusions of your choice of axioms (assumptions).
Negative physical proofs is evidence showing that a particular model (assumption) about the physical world does not hold. Negative physical proofs are relatively straightforward. Take your model, make a prediction, and then show the prediction is untrue in the physical world. This is more or less how the Bell Inequality test was supposed to be.
Positive physical proofs (for example, prove that radioactive decay is random), is in general not "actually possible". We can only show that physical reality is sufficienty compatible with your model.
Yes more or less. Bell's Theorem stated that if local realism holds (very roughly things continue existing separate from observation, and things can't influence other things faster than the speed of light), then certain measurements cannot correlate at higher than a certain rate. The relevant experiments show that for increasingly well designed and controlled cases, that these measurements correlate above the limit, and therefore local realism cannot be true.
Physical proof is not the same as mathematical proof. It is strong evidence for a mathematical model to be an adequate description of reality.
But that means we did not proof that the mathematical model is reality. There might be some weird edge case looming around the corner under which our model utterly fails to describe the physical world. Then we have to adjust the model or find a new one.
> There can be no proof, in the mathematical sense, of anything about the physical world[...]
followed by:
> Semi-related is this year's Nobel Prize for experiments showing Bell's Theorem is physically true.
The first quote refers to "mathematical proof", the second to "physical proof". They are different categories.