How to select a face created with edges? I created a mesh with Cell Fractured. I then cleaned it up adding \ removing edges , vertices, faces. But now I have these black lines when I'm out of edit mode. What does it mean and how come I don't see the black edges across the entire mesh?

Also when I try to select each individual "triangle" I can't .. it selects the entire face. I tried removing doubles, but it doesn't fix it.
Those black lines are edges which are co-planar to that face, but not associated with it. If you select those edges, they can be deleted without deleting the face. If you want to select those face sections individually, you'll have to remake them first. Start by deleting the face. Then select the edges (or verts) around one of the desired faces, then press 'F' to generate a new face there. Hope this helps!