Hairy ball theorem: Do the poles have to be opposite each other?

It is not necessary. For an extreme counterexample, see this image from the Wikipedia article for the hairy ball theorem:

This is a picture of a hairy ball where the hairs only vanish in one point — but at the cost of making the behavior in a neighborhood of that point more complicated.

The arrangement is arbitrary; the "cowlicks" need not be on opposite poles. If you want a nice step-by-step proof/exercise, see Pugh's Book Real Mathematical Analysis.

Moreover, there need not be more than one cowlick in the statement of the assertion - see wikipedia for instance.