For $\mathbf{P}^n \to \mathbf{P}^1$: Do you know the theorem on morphisms to projective space, [Hartshorne], Theorem II.7.1? Alternatively, one could use the "Projective Dimension Theorem", [Hartshorne], Theorem I.7.2.

For $\mathbf{P}^1 \to C$: Use the Hurwitz formula, [Hartshorne], Corollary IV.2.4.

For the last question: [Hartshorne], Theorem I.7.7.