Does there exists non-constant entire function with the following conditions?

Question

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be an analytic function. Pick out the statements for which $f$ is not necessarily constant.

1. $\operatorname{Im}(f'(z))>0$ for all $z\in \mathbb{C}$

2. $f(n)=3$ for all $n\in \mathbb{Z}$

3. $f'(0)=0$ and $|f'(z)|\leq 3$ for all $z\in \mathbb{C}$

My attempt: For (1) $f(z)=iz$ is a function which satisfies (1) but is not constant.

For (2) $f(z)=3+\sin(n\pi z)$ is also a non-constant function satisfying (2)

For (3), $f$ is entire implies $f'$ is entire. It is given that $f'$ is bounded hence $f'$ is constant but then $f'(0)=0$. Therefore, $f'=0$ and therefore $f$ is constant.

- Are my attempts correct?