The inspector does not switch to the arranger if clicking on the sections it only activates after clicking on another track and then back to arranger.
Isn’t this because you have your Track List divided and clicking on a part/event doesn’t switch focus between the two?
Perhaps if you enable the Track Selection follows Part Selection you would get the behavior you’re looking for.
Exactly.
Nope, just tested it 
Inspector still should follow the divided track list regardelss.