Problem with Angular velocities

So, could someone please tell me if I'm wrong and why?

You're wrong because you're assuming some relations between W0 and Wc that can't be true.
V=rw, as you've already shown you know. So if W0 is 1/6 of Wc, you then KNOW that W0 is measured at a radius R from the centre of the biggest gear... but I'm not really sure why you would need both of them, and frankly, I'm struggling to understand the information you're posting in regards to the question.

(for example a physics teacher saying that gravity was positive when something is going up......).

Which is something you can say in physics, because you define directions and therefore which way a value would be to be positive and which would be negative, because most values are gauge invariant and can be defined arbitrarily. Defining upwards as a positive value is the most common definition, so your teacher at the very least not wrong.

Edit: not the first one to rage about this. Darn you Corn!