Is it safe to say that standard deviation indicates how reliable the mean of some values is? Is it safe to say that standard deviation (SD) indicates how reliable the mean of some values is? Meaning, the standard deviation express how "correctly" the mean expresses the true nature of the values?
High SD == the mean is an unreliable expression of the data?
Low SD == the mean is a (fairly) reliable expression of the data?
 A: Q: Is it safe to say that standard deviation indicates how reliable the mean of some values is?
Standard deviation is one of two main factors contributing to the reliability of the population mean. This reliability is often quantified as the standard error (SE) of the mean, which is equal to the standard deviation ($\sigma$) divided by the square root of the sample size ($n$).
$SE=\frac{\sigma}{\sqrt{n}}$
In general  standard errors can be expressed differently depending on what is being done.
A: Q: Is it safe to say that standard deviation indicates how reliable the mean of some values is?
If you are comparing two normally-distributed variables on the same measurement scale then yes, you can regard the standard deviation as an indicator of how reliable the mean is--the smaller the standard deviation, the better able you are to "zero in" on the actual population mean.  You can also use the Fisher Information to do this (the larger the Fisher Information, the more reliable the mean is).
But if your variables are not normally distributed then it becomes trickier.  For unimodal distributions, the "reliability" of a population mean depends on the degree to which the distribution is symmetric.  For symmetric and unimodal (i.e. Gaussian) distributions the mean is a very useful measure of central tendency.  As a unimodal distribution becomes more skewed, the mean is increasingly sensitive to "outliers" in the direction of the skew and thus becomes less reliable.  For skewed distributions the median is a more reliable measure of central tendency.  In normal distributions the mean and median are equal.  I suppose that the difference between the mean and median might in some cases be a kind of rote measurement of the "reliability" of the mean.  This general concept is built into tests of normality like Shapiro-Wilk.
As the square root of the second central moment, the standard deviation is a measure of spread about the mean.  In normal distributions the standard deviation is independent from the mean but in skewed distributions it becomes a function of the mean.  In light of this, normal distributions are adequately described by their mean and standard deviation while skewed distributions are better described by the 5-number summary (minimum, Q1, median, Q3, maximum).
