Delete delete delete.
You just need to solve a few more prob stats questions and this will all start clicking with you. You are not formulating your questions in a way that implies the method, which means that you don't fully understand the methods.

More delete.

The StDev is blind to the symmetry, it is just a number which describes the spread of the data, not the symmetry of the spread. You can calculate the StDev for distributions which are nowhere near symmetrical, and it will still tell you what it tells you. Think about what it is and isn't telling you.

The symmetry is rooted in the Z_value, not the StDev. The Z_value is symmetrical in +/- about the mean because the Z_value is related to the normal distribution, which is symmetrical. We could use the student's T-distribution, for which we would be using the T_value, which is also symmetrical.

If we were trying to calculate the variance in our estimate of the variance of a sample, we would use the F-distribution, which is not symmetrical and the F_value is therefore not the same for the + as the - value.

The *_value is related to the distribution and generally requires calculus to find the exact values for a unique probability distribution.