You are concerned about non-normality, heteroskedasticity, and autocorrelation in your data.
The normality of errors is not an assumption of OLS (it is for MLE regression). That is, you can conclude that OLS is the best linear unbiased estimator (BLUE) without assuming normality. Nevertheless, there are a number of techniques within the context of robust regression to handle outliers and t distributed data, such as a Bayesian regression assuming t distributed errors.
Lack of heteroskedasticity and autocorrelation are required for OLS to be BLUE. However, with some adjustments, you can still use OLS coefficients in hypothesis testing (though they will no longer be BLUE). All you have to do is adjust the standard errors (or more generally the covariance matrix of the OLS parameters). With new standard errors, you can make new t statistics and run any hypothesis tests you want. Huber-White is common in regression packages. It can correct for heteroskedasticity in the errors. Newey-West errors are a subsequent development. They can correct for autocorrelation and heteroskedasticity. This is particularly important for some time series data that is common in finance. If you're already using Newey-West errors, then you can construct the t statistics and run whatever hypothesis tests you need to.
Another approach to autocorrelation is fit an ARMA model to the data. Similarly, with heteroskedasticity, you can fit a GARCH or SV model to the data.
Before I end, you also ask about the critical values of the distribution of errors. People are sometimes confused in frequentist statistics that the distribution of the parameters is not the same thing as the distribution of the errors. For instance, assume the errors are normal. The distribution that you use for hypothesis testing is the t distribution. What distribution is this? It is the distribution of the OLS coefficients.
That being said, there are a number of cases where the hypothesis testing requires different critical values. For instance, in the Augmented Dickey Fuller test you have to make adjustments because the coefficient being sufficiently different could mean that the underlying data will explode to infinity.