As far as I understand information criteria (IC) can be used to perform hyper parameter tuning/variable selection (e.g. I can use AIC to find the best $p^*$ and $q^*$ for an ARMA$(p,q)$ model) but not to select models from 'different classes': e.g. I cannot compare the AIC values of an ARIMA$(1,1,1)$ with that of an ARIMA$(1,2,1)$ because of the different order of differencing nor can I compare an ARMA with a ETS (as per point 5 here for example http://robjhyndman.com/hyndsight/aic/).
1) How exactly can one discriminate between what can be compared through IC and what cannot? (For instance, can I safely compare ARMA with GARCH?)
2) Is it not incorrect then to refer to IC as methods to perform model selection? Should we not rather refer to them as methods for variable selection?
3) If this is the case, what other methods for model selection are we left with apart from all the variations of CV?