Complex Analysis: Zeros of an analytic function

This can be done without Picard's theorem.

Hint for the first part: assume that $g$ has finitely many roots, and use Hadamard's factorization theorem to reach a contradiction.

For the second part, let $w\in\mathbb C$, and set $q(z)=p(z)+w$. Then $q$ is a polynomial, so $e^{\lambda z}-q(z)$ has infinitely many roots, therefore $e^{\lambda z}-p(z)=w$ for infinitely many $z$.