This post poses two different questions, but since both are related to a single regressor, lets treat them as one.
Distribution of regressors
Consider the usual multiple linear regression specification
$$y_i = \beta_0 + \beta_1x_{1i} +...+\beta_kx_{ki} + u_i$$
If regressors are stochastic then the distribution of $y_i$ results from the combination of the regressor's distribution and the error distribution. In the special case where all regressors follow a normal distribution, and the error term is also assumed normally distributed too, then we can conclude that the distribution of $y_i$ should also be normal. But this is a very special case. In general, recovering the distribution of $y_i$ is infeasible...
We side-step the matter by considering the distribution of $y_i$ conditional on the regressors. Our inference and conclusions from the model are conditional on the regressors (not only on the specific sample we have available, but in general on the regressors as random variables and on their joint sigma-algebra).
The stochastic assumptions are stated in terms of this conditionality: the error term has a zero-mean conditional on the regressors, its variance is constant conditional on the regressors. Exact test results are based on the assumption that the error term is distributed normally conditional on the regressors, etc.
All modern treatments of least-squares estimation do that. Maximum likelihood is mostly implemented as conditional maximum likelihood, although we usually don't say it explicitly: when we say "the joint density of the sample" in most cases, the term "joint" refers to the fact that we consider many observations, not to the joint distribution of the dependent variable and the regressors.
Theoretically, perhaps we would want to know the unconditional distribution of the dependent variable, so that we could handle the uncertainty surrounding it without the need to have specific information about its associations/covariance/causal relations with other variables... but if information is available, why not use it? Conditional inference is more focused (and so less uncertain) than unconditional inference but if it is mistaken (misspecification issues), it creates larger inaccuracies... This is the trade-off we have to live with. So don't worry about the non-normality of your regressor, it is not an issue.
Outliers
Outliers on the other hand is an issue. Many statistics (like the sample mean or the sample standard deviation) are not robust to the existence of outliers (while the median and the mean absolute deviation are, comparatively). There is a tendency to just delete outliers from the sample, based on the argument that they do not reflect "usual behavior" and so they are misleading -but not all situations validate such an argument and action.
There is a large literature regarding the issue. For example, look up "Influential Observations Analysis" where various measures have been constructed that go one step further regrading outliers and evaluate what degree of influence has an outlier on the estimation results that we have obtained from the model. I would say such a step is essential before making any decision about whether to delete or smooth outliers, or not, or factoring them in, when interpreting the results, or not.