David Speyer's answer is right on the money. Douady and Hubbard proved that a map of the sphere to itself whose postcritical finite set is finite has a unique "uniformization" as a rational map (up to conjugation by Mobius transformations). For a given degree, then, there are finitely many rational maps with all critical points fixed points, since these are postcritically finite.
I haven't thought about the second part of the question about whether one can bound the degree if there are only two non-critical fixed points. It might be possible to determine this from the branching data and the Lefschetz fixed-point formula.
