I mean the statistics of the dataset as a whole: it will not change covariance or correlation between variables; it will not affect a principal components analysis; you can still perform linear regression, though the intercept will be affected after scaling. For statistics within each variable - eg. mean, median, variance of each feature - those are directly affected by scaling.

I use check auto-correlation for the MA(p) parameter and the remaining partial auto-correlation for the AR(p) parameter. In both cases the result is reached by visual inspection of the plot. I'm commenting only from my (limited) experience, in which I usually achieve lower error metrics when searching parameters manually. Also I usually tend to priorize RMSE over MAPE when choosing the best manual model. I have never came across that suggestion that this method cannot be used if both orders are nonzero - I'll dig it further. Thanks for pointing that out.