Z statistic vs standard deviation

hi all,

having a few issues clarifying the difference between,:

z statistic vs standard deviation?


z = (observation - population mean) / sd vs z = (sample mean- hypothesized mean) / standard error?

finally - the standard error for large samples essentially goes to zero which means that z = (sample mean- hypothesized mean) [ as the denominator goes to a very small number] is that the correct way to think of it?