When you say you've verified that the faces are facing the right way, does that mean you've tried to "recalculate normals" on the mesh? I've had stuff import with errant faces like that before, would have been 2.56 era maybe, and recalculating the normals always sorted me. I did NOT have the phantom black thing issue tho, which as LuxXeon says, would seem more like a possible display/driver issue.

Core i7 950@3.02GHz | 12GB Corsair Dominator Ram@1600mHz | 2GB Geforce GTX 660

Lightwave | Blender | Marmoset | GIMP | Krita