For the question on cusps. I like to think of cusps as two curves meeting at the same end point. Like say you have a line that curves down and stops at say x=3. that point at x=3 does not have a derivative, because it is an end-point right? Cusps are essentially the same principle, but with two curves coming to the same end point.
Its easy to think of there being a horizontal tangent, but there could be a tangent at any slope at the point, rotating it around, which i believe is why it is undefined.