No, the p-value is the probability that the given data generated is as far away or farther by random chance given the null hypothesis is true. That is to say, we assume the null hypothesis to make some predictions and see if the data is a likely occurrence under those assumptions.

This is not the same as the probability that the null hypothesis is true. If that is what you want (and most of us do want this), then Bayesian methods are more appropriate though they are more complicated and more sensitive to initial assumptions.

Sure, but thats why the null hypothesis should be predicted by your theory. Then you are checking your theory. Most people could care less about whether there is a miniscule difference between groups or not.

>"The p-value is the probability that the null hypothesis is true."

No, as pointed out by jostylr this is wrong.