P-value adjustments in regression interpretation I have a basic question I haven't been able to find a particularly satisfactory answer. 
P-value adjustments (like the Bonferroni adjustment) are recognized as necessary when performing multiple analyses in a number of contexts, (pairwise comparisons, test of leverage for observations in a regression etc etc.) but I haven't been able to find any commentary on the interpretation of P-values in large regression models.
For example:
Take a model $ Y=\beta_0 + \beta_1 X_1 + ... +\beta_nX_n $
We are confident in the quality of the model (perhaps we've already run a stepwise model selection method with AIC or BIC). Now what we want to do is consider which $\beta$-values are "significant", meaning that we can confidently interpret them. Those that are not "significant" we will still include in the model, (because using a measure of model quality like AIC we've already determined we would like to include it) but we will instead consider them as variables we are controlling for. This is a common approach in many social-science methods. 
For each $\beta_n $ we can obtain a p-value $p_n$ and some significance level $\alpha$.
To determine which regressors are "significant" vs. which are "controls," should we adjust are $\alpha$ based on the number of regressors? I've seen a lot of published research that doesn't do this in my field (education research), and it seems to lead to over-stating the significance of variables. Any references for rational would be appreciated. 
As an example, see this paper published in a good educational research journal.
 A: Philosophy aside [see below], the answer to your actual question is complicated and one that people argue about a lot! Rothman (https://www.ncbi.nlm.nih.gov/pubmed/2081237) is generally against adjusting without good reason, as you increase your false negative rate. However, others argue that correction is appropriate in this instance (just as it would be in many univariate tests) - for example, see https://repository.upenn.edu/cgi/viewcontent.cgi?article=1001&context=wharton_research_scholars. 
There is a trade-off, and the appropriate answer depends on the question, and on the value applied to false positives and false negatives. That is, if rejecting the null hypothesis has low cost (e.g. follow-up in a cheap mathematical model), you would favour no correction. If rejecting the null hypothesis has higher cost (e.g. follow-up with genetically modified mice), you might favour conservatism and Bonferroni correction.  
~~ Philosophical disagreement ~~
I think your general construction is philosophically wrong here - you shouldn't be judging interpretation based on significance - that is, the fact that a coefficient is significant in your model does not mean that you can confidently interpret it. 
Determining which coefficients can be confidently interpreted is more a matter of how large you would expect the coefficient to truly be, and whether your design has sufficient power to detect that size of coefficient. That is a function of the significance level, but is only partly determined by it.
Really, you should be determining which terms are variables of interest, and which are covariates, prior to any analysis. 
