Yes, I think that airplane pilots should be required to spend more time on the ground. Pilots need to get an appropriate amount of sleep and / or rest between their flights. I tired or jet lagged pilot is not only dangerous to himself, but also to those that he transports and the people on the ground.
No, airplane pilots should not be required to spend more time on the ground, because there are times that they just want to finish their last flight, but it ends up taking them longer, because they have mandatory rest periods first. Pilots are professionals, and they should be allowed to decide when they are too tired to fly.
No, I think there is no real evidence that this would be beneficial to the safety of passengers. They already have very strict rules about how many hours they can work, and work far fewer hours then people in other demanding jobs such as surgeons, etc. They also need to fly enough to keep their skills up.
I do not believe airplane pilots should be required to spend more time on the ground. Just like anyone else they aren't getting paid if they aren't working and it's been made commonly known that airplane pilots don't make as much as one would think. Let them fly, they know what they're doing and what they can tolerate.
I think that it is up to the airplane pilot if he/she feels that they are capable of flying the plane in their current state, the vast majority of them are smart enough to stop themselves if they feel they are to tired or could not handle it, I don't think more rules and regulations are required on them.