Normal distribution necessary to assess moderating and mediating effects? 
*

*Is linear regression only suitable for variables with a normal
distribution?

*If so, is there an alternative nonparametric test to test mediation or moderation? 

 A: For those not familiar with the language, moderation and mediation were both discussed in Barron and Kenny's influential article (free pdf).
Mediation
With regards to mediation, bootstrapping is often used where normality does not seem like a reasonable assumption.


*

*For SPSS and perhaps other implementations, check out the macros on the website of Andrew Hayes

*For R  have a look at the mediation package
Moderation
With regard to moderator regression, you could also explore bootstrapping options if you were concerned with the accuracy of the p-values you were obtaining.
A: Many of us use linear regression in rough-and-ready fashion to learn about the relative importance of predictors, to assess the shape of relationships, and so on.  But if one wants to make strict probabilistic inferences one needs to satisfy the set of standard assumptions entailed in such regression.  The most important of these are random sampling and independent observations.  Another one is that the regression residuals (not any particular X or Y variable) should be normally distributed.  Without satisfying all of the standard assumptions, one cannot vouch for the accuracy of standard errors or p-values.
