The empirical rule for a normal distribution suggests that 68% of data will fall within the first standard deviation, 95% within the first two standard deviations, and 99.7% will fall within the first three standard deviations of the mean. If I have a data set where all the elements fall within first two standard deviations of the mean, can I consider this data set as a normally distributed one?
The naive approach would be to expect 5 samples of 100 outside 2 SD, with a standard deviation on that of $\sqrt 5 \approx 2.2$, so getting 0 is only a 2 SD event (and maybe not that bad as the measured SD is probably a bit low).
A normally distributed random variable has a pdf with support over all of $ \mathbb R$. If it does not it is some sort of truncated distribution. Check the properties of a truncated normal distribution on Wikipedia to verify if your data satisfies those.