I try to answer following question. I would like to find a general approach for part b, since I think any entire function with $n $ roots is either polynomial of degree $n$ or is a product of polynomial with another entire function with no roots.
Give an example of an entire function that is not a polynomial hand has
a) no root;
b) exactly $n$ roots;
c) infinitely many roots.
For part a) I think $e^z$ works.
Part b) I think $f(z)=e^zP(z)$, where $P(z)$ is a polynomial of degree $n$. However,I would like to know if there is any entire function not constructed by any polynomial.
For part c) I think $g(z)=\sin(z)$, which $z=n\pi$, where $n$ is an integer.
Your answers are fine. As for your separate question: let $f$ be entire with only $n$ roots $z_i, i=1\ldots n$. If $\alpha_i$ is the order of the root $z_i$, then $$g(z)=\frac{f(z)}{\prod_{i=1}^n (z-z_i)^{\alpha_i}}$$ is an entire function with no roots. Therefore $f$ is the product of a polynomial and an entire function with no roots.