Does ADF test regression remove all autcorrelation? (Do we really know with 100% certainty that ADF-resulted-regression has no autocorrelation?) The goal of ADF is to test whether we have unit root or not. 
DF (Dickey-Fuller) test equation (regression equation) may include autocorrelation, and its result is not so reliable.
In ADF, additional lags of the differenced variable are added to the right of the ADF regression equation. These added lags of the differenced variable reduces autocorrelation.
Now, assume that we are performing ADF test when max lag length (say 14) is given (or assume max lag length (say 14) is the obtained from Schwetz's formula; as in Eviews).
If ADF test results in optimal lag length of 5 (via AIC or ABC criterion) in ADF test regression equation, then do we really know that the ADF equation with 5 lag length includes no autocorrelation?
OR, do we have to check that the ADF test regression equation with optimal lag length of 5 includes no autocorrelation by the help of some autocorrelation tests?
Any answer with related sources will be greatly appreciated.
 A: ...then do we really know that the ADF equation with 5 lag length includes no autocorrelation?
No, we don't. Lag length selection due to information criteria such as AIC or BIC does not guarantee that there will be no autocorrelation in model residuals. 
AIC and BIC are there to solve the problem of finding a well-fitting (in terms of high likelihood) model without introducing too much variance due to trying to estimate a large number of parameters. Thus they help you strike the balance and find a parsimonious model. They do not explicitly help avoid autocorrelated errors but they may do so implicitly to some degree by trying to maximize likelihood (subject to the penalty for the number of parameters).
...do we have to check that the ADF test regression equation with optimal lag length of 5 includes no autocorrelation by the help of some autocorrelation tests?
If you are particularly concerned about autocorrelated residuals, you will have to use some tests to check if autocorrelation is there. But when you think about it, you might as well be willing to accept some (relatively small) degree of autocorrelation as long as estimation of a richer model (a model with more parameters) that removes the autocorrelation yields a relatively poor result due to the increase in variance associated with estimating the extra parameters.
