**Geometric proof**:
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 $\mathbf{P}^n \to C$: There is a $k = 2$ or $3$ such that there exists a closed immersion $C \hookrightarrow \mathbf{P}^k$. Similarly to the above, there are no non-constant morphisms $\mathbf{P}^n \to \mathbf{P}^k$ for $n > k$.

**Cohomological proof**:
Alternatively: If $f: X \to Y$ is surjective, $f^\ast: H^\ast(Y) \hookrightarrow H^\ast(X)$ is injective for a Weil cohomology theory $H^\ast(-)$. For $\ell$-adic cohomology, $H^*_\ell(\mathbf{P}^n)$ can be easily calculated: $H_\ell^{2i}(\mathbf{P}^n_{\bar{k}}) = \mathbf{Q}_\ell$ for $i = 0, 1, \ldots, n$, and the other cohomology groups are $0$; and $H^1_\ell(C_{\bar{k}}) = \mathbf{Q}_\ell^{2g}$, $g$ the genus of $C$.

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