FAA has no sector limitations. If you're doing 20 minute legs you could do 24 a day.
Change of date doesn't matter. It's per duty period or really between rest periods. Varies some between domestic and internatinal and charter but the basic domestic two pilot crew can fly 7+59 before needing a minimum of 9 hours of rest per the regulations. (Most airline contracts are more restrictive.) Say you come on duty at 23:00 and depart at 00:00. Fly 5 hours, :30 on the ground and the next leg is 3 hours. Add 30 minutes again for a post flight duty period and it's 09:00. Now it starts getting tricky because you're going to go over 8 in 24. You are required to have 11 consecutive hours rest if your going to fly more than 9 in 24. So you check in again at 20:00, launch at 21:00 and fly past 00:00. So you've legally hit 11 hours in 24.
No, the rest must by at least 9 hours consectutive for less than 8 in 24, 10 hours for less than 9 and 11 hours if they schedule you for more than 9 in 24. It's really a look back, at the end of the flight you have to be able to show where you had the required rest in the prior 24.