I suggest you look at this thread. This is a '36 Ford pickup with a chopped top that got a new headliner: Custom Headliner for '36 Ford Pickup
To answer your questions: This is a "hard" headliner, but is finished off just like a regular headliner around the perimeter. It needs to be made from something stretchy like vinyl or Ultraleather.
1) PVC foamboard, which cuts and shapes with regular woodworking tools or a utility knife. If you drill large holes, do them with spade bits or Forstner bits, and go slowly.
2) 1/8" (3MM) is plenty
3) It comes in two main thicknesses, 3MM and 6MM thick and in 4' by 8' sheets. (I found a supplier that has it in 5' by 10' sheets) You won't find anything you need at Home Depot or any other home improvement store other than 1/4" luan plywood if you decide to make a base for the foamboard. PVC foamboard is available from most sign shops and from sheet plastic suppliers. It goes by the brand names Komatex, CelTec, Sintra, and Palight. A 4' by 8' sheet of 3MM foamboard sells for about $30. ABS is very expensive and a lot harder to work with.