1) You say "I am aware that box-cox transformation MAY make data set significantly normal distributed with constant mean and variance."

No  it simply decouples a possible relationship between the variance of the errors and the Expected value. Please read my answer to https://stats.stackexchange.com/questions/18844/when-and-why-should-you-take-the-log-of-a-distribution-of-numbers. It MAY generate a more normal distribution if the series is leptokurtric (fat tails)


2) You say/ask  "does it is always make variance and mean constant?"

Not necessarily as the error variance may change deterministically over time requiring Generalized Least Squares (Weighted least Squares see http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html ) or stochastically over time requiring a GARCH add-on.



The reason Box-Cox shows up is that in the "old bad days ! " that is all that was known or in textbooks as a way to treat heteroscedastic errors. Textbooks are out of date as soon as they get printed. SE is the true textbook because it is ever evolving to improve the craft .


Also see a number of my posts where the error variance changes and in particular https://stats.stackexchange.com/questions/384905/removing-variance-in-time-series-after-applying-log-transformation/384907#384907 might be of interest.



EDITED after remarks  

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