1..Rigorous is the term most often used when talking about mathematical proofs. It is also used when talking about any discipline that requires very precise obedience to rules, or something that is very difficult. " A rigorous course in Latin, a rigorous hike, a rigorous physical workout.."
Strict would be used when talking about parents, or a teacher who demand obedience.
2. "She is strict" is more natural
3. Her final exam grading is rigorous is more natural, but in the case either would be acceptable.