It was mentioned in class that checking the inner products with the normals would work as long as the winding of the edges was consistent. In practice, is there a preference between winding clockwise and counterclockwise?

I think either is probably ok as long as you know how you define normal and are consistent. If you go clockwise ( from P1 to P0 as the tangent vector), then your Normal points inwards. The "inside" of the triangle is then the region where the dot product of V and N is positive. But I'm new to CG so perhaps there's a convention all computer graphics people use.

Interestingly, the default triangle winding used in OpenGL and WebGL rendering libraries (e.g. to determine whether a triangle is facing forwards or backwards) is counter-clockwise. This results in a triangle being front-facing (the normal vector points towards you) if the vertices wind counter-clockwise. This follows the right-hand rule used in mathematics and physics!