Is this possible? To take an edge, and extrude them not along some axis, like x, y, z, but along the axis of the original sloped face.
OK, I will suppose that you mean extruding an edge while remaining in the same plane as the face from which it is a part.
You just have to make a CTO of that face (shortcut Ctrl-Alt-C) which will become your secondary transformation orientation. Then the easiest to get used to it will be to use the transformation widget to guide your extrusion, at the beginning.
What does CTO mean? Can you explain a bit further?
CTO = custom transform orientation
It is an orientation for your transformation, like are the default TOs ‘Global’, ‘Local’, ‘Normal’ and ‘View’, except that they are your creation.
Select an object or any element(s) of the geometry of that object to create you CTO using the Ctrl-Shift-C. As I said, the transformation widget is useful to figure out how the resulting CTO is oriented.