Keep in mind that the MLE of the variance divides by n, not n-1 so your sample data in view of the MLEs don't have equal mean and variance. But since you have additional information about the relationship between the mean and the variance one shouldn't expect that the sample mean ends up being the MLE. If the data behaves as described then it should be close but not necessarily equal.

Also you can use the info you weren't sure why they gave you for part (b). That's why they gave you that info.