Can you calculate (or estimate) standard deviation using mean, median, and sample size?

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If you know the mean, median and sample size (with no other information), can you estimate standard deviation? Or, let's say you know one value from the sample (only 1), can you tell which percentile that value is based on this information alone?

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No. You could draw $N$ samples all with the same value $x$, giving a mean and median of $x$, or you could draw $N$ samples normally distributed around $x$, also giving a mean and median of $x$, but these are totally different distributions. However, if all you know is mean and median you won't be able to tell them apart, because these quantities are the same for both sample sets. Knowing the value of one sample doesn't change this (all that this could possibly tell you is that there might be at least multiple different values in your sample set, if the value you know is different from the mean and median).