Z - score

Well, my question may seem a little silly to most advanced in the area, but it has taken my sleep. That is the question:

I have a set of continuous quantitative values and such values do not follow a normal distribution.

After turning my data to one logarithmic scale in base "e" (neperianos), I realized that my data obtained a very close distribution of normal.

Shortly after these transformed values of logarithmic scale were transformed again to Z score values.

Is it right to do this? I also thought he could turn to score Z- directly the raw values. Without the need to turn logarithm before.

However, by applying the Z-score distribution to amounts previously transformed to logarithms, I realized that my timing was perfectly normal and standardized. (Mean = 0 and standard deviation = 1). The Histogram was very good.

After much reading and studying, I realized that some tutorials and handouts say we should transform values of a normal distribution for Z-score.

That is, if the distribution is not normal we can not turn directly to z-score?

Sorry for the long text. If anyone can make me doubt this will be very grateful.

The last question is: I thought that Z - score or normalization is not the same thing as normal distribution. Am i right?