It's been a few decades for me but, SS trip units get their control power from the current sensors.I believe there is a shunt in the CT circuit that creates a voltage to power the trip unit. Some trip units can take a separate power source for fault memory and metering data.
The tests generally performed in the field are single phase. IIRC the time tolerance bands are much wider that what we would expect from a digital relay used in substations.
Like anything mechanical, if left in the same state for many years, a delay in operating can be expected. If its a LCPCB, the mechanism is somewhat exposed, so depending on the environment things can get dirty and gum up. NETA publishes a frequency of inspection / test chart that should be available free to download.
Most manufacturers will sell a secondary test box to check the trip unit. NETA will call for primary injection testing. The secondary box is the size of a suitcase-operates from 120V source. A primary injection test set weighs several thousand pounds and requires 480V (preferably) or 240 with an autotransformer. Back in the day, there was some concern about the firing angle of the primary sets, which could cause erroneous results, especially for INST tests. I believe most "modern" primary sets have better controls to mitigate these issues.
In my experience years ago, it was not uncommon to find a solid state trip unit that failed in some way during maintenance testing. There are several reputable firms that specialize in trip unit repairs. I'm fairly certain that Microprocessor based trips units have some kind of alarm contact that can be connected to SCADA or annunciator.