My colleagues and I are having discussion whether it's valid to calculate standard deviation for $n=2$ or not? I think it's valid since I can calculate based on the equation, but higher N will give more power in the analysis. Can anyone comment? Thank you!
Standard deviation is a measure of spread from the mean, so it is defined even when $N=1$ (although in that case it will always be 0). Certainly when $N=2$, it is a meaningful statistic.
And you are right -- if $N$ is larger, the statistic will be more powerful.
It depends on how many "degrees of freedom" you have, or how much data you have left after estimating the other parameters you need before getting to the variance. If you have a single univariate sample and just want to estimate a variance, you can since this only requires an estimate of the mean. If on the other hand you have a simple linear regression and have fit an intercept and slope, then you have no degrees of freedom left and can't estimate the variance of the error term.