For linear regression, why do people usually standardize the X variables and log transform Y variables to make them normally distributed? In many Kaggle competitions where linear regression has been applied, I see people plot the y distribution and then take the log of (or other transformation of) the dependent variable to make y normal distributed. What would happen if we don't do this step: transform the dependent variable into a normal distribution?
I have a few thoughts but wasn't sure which one is more correct and which one is wrong.

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*if y is skewed, after running the linear regression model, we could have the heteroscedasticity problem (variance is not constant along the predicted y)

*we know the linear regression coefficients are still unbiased even if y is skewed; however, the t-test of the coefficient wouldn't make sense anymore because y is not a normal distribution.

*Because X is usually standardized, y's distribution should match X's normal distribution.


All 3 of your points are incorrect.
The log transformation does not necessarily make the dependent variable normally distributed. But more importantly, it doesn't matter if the dependent variable is normally distributed or not - or the independent variable either. Instead, what we typically care about is that the residuals of the fitted model are normally distributed, and even that is a helpful feature, not a strict requirement. Also, the residuals can be normally distributed even when the predictor and response are not.
You may find these threads helpful:
When (and why) should you take the log of a distribution (of numbers)?
Interpretation of log transformed predictor and/or response
