I read something about standard error, which tells that sample mean is not accurate estimation because we do not sample full population of size N. But, what if sample size n = N or exceeds N, i.e. n > N? Can standard error can be more accurate than the standard deviation?
Assuming that we are talking about sampling without replacement, if the sample size exceeds the population size, you need to rethink your stipulated "population size". When $n=N$ in this case you have sampled the entire population (what we call a "census") and so all measurable descriptive quantities relating to the population should be known. Sensible estimators of such quantities will have zero standard error in this case.