One post has caught my attention (https://support.bioconductor.org/p/35159/), which highlights how to "skip" the eBayes function. First, I understand that Bayes methodology is why limma is so popular and the benefits of utilizing such methods. But if one calculated the t-values and p-values as specified (code below), is there anything that distinguishes the results from ordinary least square if lmFit(method='ls") is used? In other words, if I wanted to compare empirical bayes results  to ordinary least square results, would this be an appropriate way to examine the differences?

How to calculate t-value and p-values manually:

fit2$t <- fit2$coef/fit2$stdev.unscaled/fit2$sigma
fit2$p.value <- 2 * pt(-abs(fit2$t), df = fit2$df.residual)

Yes, you can do a regular t-test using the code you give instead of an empirical Bayes moderated t-test. The code you give is from page 61 of the limma User's Guide. Yes, the code will give you same results that you would get from any statistical package able to fit a similar linear model.

Just a note on terminology though. limma always does least squares (either weighted or unweighted) and empirical Bayes doesn't change that. It is at the test statistic stage that the empirical Bayes comes in.