There is in fact a probability attached to p-values. A p-value of 0.05 for instance means your conclusions will be wrong 5 out of 100 times. You can reduce the p-value to e.g. 0.001 or any other value you want.
No, it means that the probability of seeing an effect of that magnitude on a dataset of that size when the null hypothesis is true will happen due to random chance 5 out of 100 times. It says NOTHING about your hypothesis, it is entirely a statement about the null hypothesis.