Choice of replacing missing values based on the data distribution
I would not definitely recommend substituting missing values by mean or by median or mode. If you want to go through some techniques and get a glance at them, I would recommend going through this link and for imputation techniques this wiki page gives you a brief .
Do you think that there is a way to predict missing values from the other cells. If yes, apply a regression model on those variables and estimate the missing value. But remember this lacks variability as values fall on the regression line itself. There are methods like regression imputation which can add this variability component to the estimated value.
If you are unable to go anywhere from the previous step, then see how the values are distributed for the missing variable, substitute them according to that distribution by using a random function.
And if your unable to perform any of the above mentioned ones and want to go by mean, median. I cant really give my opinion as in this case they are nearer to each other. See what gives you the best predictable and decide between them.
Coming to your final question, Std. deviation just shows how far your values are falling away from mean. If your data has large range with good enough number of  points distributed at extremes, you would be expected to have high std. deviation.