Depends on the airline. We stop for 5 main items:
1. Any fire
2. Engine failure (identified by 2 or more parameters)
3. Configuration warning
4. "Monitor radar display" warning
5. "Windshear ahead" warning
Answer 1: Yes
Answer 2: They don't need to as the very nature of the configuration warning means the aircraft is unsafe or unable to fly. In which case they have covered themselves with their fourth reason to reject between 80kts and V1.