some tips for 1.5 carol story.
1. quest tips say "go to the school's park", but this requires you've started carol story, so ignore it for now.
2. the exam happens on the last day of the month. you can know which day it is at the top right corner. this bit took me few hours because the 31th of my game happens to be a sunday. i thought i missed something so i loaded a previous save and replayed few hours. if you can't be bothered to skip days by hand, use mods.
3. once you pass the exam, when you talk to carol in the library she finally says something different than "knowledge is the greatest weapon for ignorance". talk to her a few times, quest girls tips will say "continue visiting carol", you should keep visiting her in the library until a friday, because story progresses automatically on the following friday. so don't skip school on friday.
4. after carol becomes your girlfriend, the story will progress automatically on the next monday and then quest girls tips will say "go to the garage and speak with carol".