Dividing any number by zero produces a result which is just not defined (in most situations) so it means nothing here. 0/0 would mean there is no standard deviation. You can have a situation in which you the data-point is the same as the mean, so when standardizing you'll have 0 divided by a standard deviation, which will give you 0 (since you can divide 0 by any number and it will give you 0).  A z-score of zero is fine, its right on the middle of a normal curve--the 50th percentile.

For repeated entries, if you mean that the same value appears multiple times, then I can't think of what a problem might be. If by repeated you mean the data points are contingent (not-independent from each other) then you can run into problems with later statistical treatment...not totally sure what else