Problem about estimation of median and chi-square test for normal distribution

Hi guys, I have a problem on estimating the median of a population and I am not sure how to do so.
I am not sure not to estimate the median of the travel time.
1. Is the median of the sample a unbiased estimator of the population median? If not, how am I supposed to find it?
2. I have tried to test that X & Y are normally distributed using chi-square test but I am confused about how should I group the categories?
From my understanding, I have to group the them so that the expected frequency of the group >=5. But when I try to do this. I find that there are only 2 groups. Then the degree of freedom = 2(groups) - 2 (cause estimate of mean & variance is used), which is incorrect. Where is the problem?

Thank you in advance!