You can indeed compute means and standard distributions for any distribution with small enough tails, but if the distribution is far from normal then they may not be very useful statistics. E.g., an important reason why the mean of a bunch of samples is an interesting statistic is that if the underlying distribution is normal then the sample mean is the maximum-likelihood estimator of the distribution's mean. But, e.g., if the underlying distribution is a double exponential then the max-likelihood estimator for its position is the median rather than the mean. Or if the distribution is Cauchy then the sample mean is just as noisy as a single sample.