With AFP and AFP Deluxe:

-- Can I place myself in the Pit Area and fly as far away as I wish?
Sure can.

-- Can I trail behind the plane and fly beyond the field boundary? Would I see any terrain?
Yes you can, but only without terrain. It will show you the planes position in the 2nd window top right.

-- Can I fly over the pit area (or does it reset the plane to do so)?
Sure can. There are no boudries on the field you fly your real planes, why should there be on the sim? You need to learn how to react when you get near the pits, not wish there was a wall.

Two other questions if you can help, please:
-- If I crash, can I control when to reset the plane (or does it do so automatically)?
Yes you can. AFP and AFPD gives you the option to time reset from 0-1 minute. You can also get into the program and make the timer as long as you want. Or you can use the spacebar to reset. This is good if one of the control surfaces break off AFP and AFPD will detect that the control surface is missing and will fly as though it would on the field. This feature is really cool, because it shows you that it is possible to land a plane with only one wing. Takes some practice, but can be done.

-- Can I edit planes to not only behave differently but also look different (repaints don't count)?
AFP and AFPD have almost a countless amount of parameters that you can change. Much more then you can do in real life like being able to control the prop wash factor and dozens of other parmeters.