Do I need to test for autocorrelation or normality assumption if I am running the regression with standard errors? I used OLS regression to estimate a relationship between X and Y with a couple of control variables. However, when I tested for heteroskedasticity with Breusch–Pagan/Cook–Weisberg test, my residuals were heteroskedastic. I tried log transformation for my dependent variable, but it did not resolve the issue.
My data (950k observations) has outliers as per boxplots, histograms, and extremes test(in Stata), but they are not actually outliers. Those are valid observations, but a minority in my large dataset. Hence, I don't think winsorization was an ideal approach. I finally resorted to regression with robust errors in Stata.
Since the problem of heteroskedasticity is no longer a concern with robust errors, do I still have to test for autocorrelation and normality of errors (the other two assumptions for OLS)? Why or why not?
Secondly, is reporting regression analysis with robust errors okay in research papers?
Thank you. Appreciate any help.

With so many observations, tests for normality or autocorrelation will most likely end up giving extremely low $p$-values, suggesting to reject the null.
Using robust standard errors is fine and perfectly acceptable. Perhaps you may consider doing a bit of model selection (say, lasso, stepwise, subset regression, etc.) if you are uncertain as to which variables are relevant and which are not.
