stability step response
2. In various applications, system behaviour in time domain is the primary quality criterion, e.g. with pulse amplifiers or control theory. It's of course related to small signal frequency domain representation but also involving large signal effects.
3. As far as I know, mainly empirical parameter optimization according to a defined quality criterion is applied in time domain, in some cases also simplified tuning methods, e.g. ziegler-nichols for PID controllers.