For #15 of the Dynamics part of the AP style problems, why can't we use mgsin instead of Normal force, AND why can't we use mgcos for Normal force - why do we have to derive another one.

I'm not sure I completely understand the question, but if I attempt to read between the lines, you're wondering why this isn't the same as a block sliding down a ramp.  Recall for the box sliding down a ramp scenario, we're setting the motion of the box along the x-axis and 'tipping' our axes to line up with that motion.  In this case, although it looks like a box on the ramp, the car is actually moving into or out of the page, not sliding down the ramp.  The direction of its motion is centripetal as it goes around the curve, or directly to the right (along the x-axis)... so we don't tilt our axis in this problem.  As the car's motion is along the positive axis, directly horizontal, that remains our 'reference' axis.