No the variance (for a univariate statistic or RV) is the mean of the square difference between the data and the mean and the square difference is always positive so the variance is always positice.

2. Can you use the empirical rule/Z table if your mean is 0 and your standard deviation is 2?

For some reason I thought your mean had to be 0 and standard dev had to be 1.

You should be working with z-scores, the SD of the z-score is always 1, as you divide the deviation from the mean by the SD of your raw statistic.

RonL