It is true; the z value has no unit of measurement. The formula for z is: (x-mu)/sigma and the units in the numerator and denominator cancel out.
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It is true. Because the normal distribution is above the horizontal axis for all values, the area under it is a positive quantity no matter the z value.
A z-chart in statistics is a chart that contains the values that represent the areas under the standard normal curve for the values between 0 and the relative Z-score.
No, they do not. They are pure numbers.
Z score of 0 is the mean of the distribution.
It is true because the distribution is symmetrical about Z=0.