1. Yep, any character with an exclamation mark has an active quest that needs to be done (more importantly, any character with these flags will also basically be IMPOSSIBLE to interact with 'normally', so no giving them Corrupters and mind controlling them to love you, gotta complete quest first). The vast majority of characters with quests will require you to go to the Private Investigator in Downtown and hire him for around $100+ a day to look into them (this does not apply to all, "Shimekaze", Hooters, and some other characters will have different quest starts... I'm not listing them all here).
2. People have mentioned the guide and such, but for a semi-useful one in-game you can check your phone in the bottom left. That will have notes regarding important PTA members or other staff members. For hard mode, if you know what you're looking for but not sure how to progress, you can try to view the event stuff in the Debug in the bottom right, though I don't recommend it unless you already know of an event but want to know why it isn't kicking off (i.e. stuff like "oh I need to do this X amount of times when I see this character here" kind of events where it always happens in the same spot, or you can try to look at 'person attached events' which are tied to a character but not a location specifically)
3. The guide is a bit outdated, but still is the best source for a one-stop shop to look. Alternative is either asking here, or on the "new official" forum site, I'm not aware of an active wiki.
*quick tip: I personally investigate June Perrier asap/day 1, the Saturday tutor stuff you get while doing it is an early source of money (especially with nerfs to working at the diner) and it's also one of the few ways to raise your reputation outside of specific events at school.