The conditions that must be met when using the standard deviation formula when conducting a significance test are that the samples should be independent of each other and the variance is known and not infinite.
The difference between p and p-hat is that p represents that true standard deviation of the population while p-hat is only an approximation using a sample of the population.
The conditions that must be met when approximating a normal distribution are that the samples must be independent of each other and that number of samples must be at least 30.