During, yes, sure..
After, no
FDP ends at IN time
FAR Rest starts at IN time + :01
Yes, contractual duty period is :15 after IN
Yes, rsvs are required to check for follow-on assignments... There is a whole set of flowcharts for that, AA / APA are positively addicted to complexity
They can notify the crew during flight, but not during sterile period. (No, I don't know how they know, but I do know how I know...)