Answer 1 : The reason that Mincer used lny as the dependent variable instead of y(earnings) is estimating the returns of scooling and tenure. you can find this logic in textbook of labor economics such as a chapter of human capital. So you must lny as the dependent variable, not y.

Answer 2 : Mincerian earnings equation is the above that you take correctly. In the equation, the 'u' is just a residual of OLS estimation. So you don't worry. You never include 'u' in the regression command of Stata as below :   

you just run stata regression command as follows:

reg lny s e e2

which 'reg' is regression(OLS, Ordinary Least Square), 'lny' is log of y(earnings), 's' is the shooling years, 'e' is the tenure years, and 'e2' is the square of 'e'.

Before running the above command, you should make two new variable such as lny and e2.

Also you just run like

gen lny = ln(y)

gen e2=e^2

I hope you get your answer.

thanks a lot