Yes, this is normal. An assumption of the MNL model is that the random component of utility is IID and takes a Gumbel distribution, often called the Generalized Extreme Value distribution Type 1.
Regarding your utilities, you should be aware that when using HB estimation, there is a difference between the so called "upper model" and the "lower model". On the upper level you assume that the utilities are distributed multivariate normal. By contrast, on the lower level, you use individual observations together with your upper level distribution to get individual estimates. This can result in distrubutions that are not normally distributed.
Regarding importances you whould be aware that these estimates do not incorporate differences in scale. You should also take into account the error terms when comparing individual parameter estimations. So you can use a t-test, but I think you are better of using another model, e.g. latent class mixed logit.