ARIMA models : plot_diagnostics, what's meaning of residuals of our model

Residual indeed is the difference between true and predicted value. If there are correlations between residuals - there is information left in the residuals which should be used in computing forecasts. If the residuals have a mean other than zero, then the forecasts are biased. For instance if we have a constantly growing residual like (... -0.3, -0.2, 0.1, 0, 0.1, 0.2, 0.3, ... and so on, the mean will be 0) it means that our model does not fully depict the process.

Parameters: If you look at the package documentation you will see that these parameters are used to ENFORCE stationarity or invertibility. If the data is stationary and the AR parameters are chosen correctly (since you should have done some previous data preprocessing) why should we do it again? Same stands for invertibility.