I think it's worth remembering that this game is intended as a toolkit for a variety of potential stories. The rule book doesn't explicitly answer your question of what the investigators look into and why, but that's because the answer will depend on the mystery that the GM runs and the characters that the players create. Some games are made for very specific types of characters with very specific motivations, but in my experience Eureka is more focused on creating characters with all of the messy situations and motives of real people and then seeing what happens.
(In terms of the other questions, I think the game makes the motivations pretty clear: playing as a cop would make the game less interactive, because it would necessitate spending more time on police procedure and less on the characters personally interacting with the case, and monsters usually prefer to hide their true nature because it's scary and causes harm to other people.)