When should we apply normality tests? For which types of the variables should we apply the normality test? For example dependent variables, independent variables, or control variables, etc?
In many contexts (e.g., multiple regression) it is the distribution of the residuals that is the relevant assumption (see here for a discussion).
Your question also raises the general issue of the relevance of normality testing. See this previous question: Is normality testing 'essentially useless'?
However, more broadly, doing exploratory data analysis (e.g., plots, summary statistics, etc.) to understand the distribution of all your variables is important.