Independent variables in multiple linear regression I have a set of experimental parameters and my task it to find reasonable descriptors to describe them (chemistry).
Since I've got descriptors, I checked Pearson correlation for each of experimental parameter and then chose descriptors with P<0.05 for further investigation. But at this step I did not check normality of my data. After that I proceeded with multiple linear regression and found suitable models. For each model I checked normality by Kolmogorov-Smirnov test with Lilliefors correction and also by Shapiro-Wilk test (o.5 from 2). All models passed these tests, however visual diagnostics of normal Q-Q plots showed some non-normal distribution in my opinion. So, my questions are:
(1) Is it enough to rely only on Kolmogorov-Smirnov and Shapiro-Wilk tests to check normality of models?
(2) Is it correct that I did not check normality of my descriptors before doing MLR or it is a serious mistake? If it's not OK, does it mean that I have to transform non-normal distributed descriptors and then repeat Pearson correlation once more?
(3) Say, I have non-normal distribution for one descriptor in model, whereas two others do not have such problem. Can I do transformation only for this problematic descriptor or it should be done for all of them? I supposed it is very simple question and answer is yes, but anyway. 
 A: (1) Use these tests and various graphs for residuals from the models which you have done 
(2) I think that it is normality of residuals which is of more interest. Various test statistics are valid if results hold. I do not think it is much use to transform predictors just because you want their distribution to resemble normal distribution. Usually variable transforms are done because you might have problem with heteroskedasticity, which is non constant variance of residuals. 
(3) As in previous question, check residual normality and transform only if you have problem with heteroskedasticity.
A: First, normality of model residuals is not needed for OLS estimates to be BLUE (best linear unbiased estimators, where best means minimum variance) asymptotically. Unless your sample is small (where asymptotic results do not kick in yet), non-normality is not a problem.
Second, selecting descriptors based on correlations with the response variable need not be a good strategy. Often descriptors are jointly significant when they are not significant individually. You may want to study strategies for selection of descriptors more before proceeding (look for tags feature-selection, model-selection, regression-strategies).
Now regarding your questions:
(1) I do not have a strong opinion. These tests should be OK.
(2) Descriptors need not be normal for OLS estimates to be BLUE. Non-normality of descriptors is not a problem. For example, dummy variables are frequently used in multiple linear regression, and clearly they are not normal.
(3) As I said in (2), non-normality of descriptors is not a problem.
