First, having normally distributed errors is equivalent to having normally distributed observations for any linear time series model.
Second, it is not necessary to assume normality of errors. Often, maximum likelihood is used to estimate the parameters of the model, and then a Gaussian likelihood is used, but it gives good results even with non-normal data. Where normality of errors is often assumed is in using the AIC for order selection, and in computing prediction intervals.
There are several specifications of ARIMA models with exogenous variables, and more than one such specification has been called an ARIMAX model, so it is not possible to precisely answer your second question without you specifying the model more accurately. For discussion of some of the models, see http://robjhyndman.com/hyndsight/arimax/