You have to know some (basic) facts about complex/holomorphic line bundles over complex manifolds. I'll try to be much clear as possible.
(1) The first Chern class is a topological invariant for complex line bundles in the sense that it gives a group homomorphism $c_1:H^1(X,\mathcal{O}^*_X)\to H^2(X,\mathbb{Z})$. So we can think of $H^2(X,\mathbb{Z})$ as the group parametrizing complex line bundles.
(2) Holomorphic line bundles are parametrized by $H^1(X,\mathcal{O}^*_X)$
(3) There exists a long exact sequence of cohomology group, which the relevant part (for our purpose) is the following \begin{equation*}\cdots\to H^1(X,\mathcal{O}_X)\to H^1(X,\mathcal{O}_X^*)\overset{c_1}{\to}H^2(X,\mathbb{Z})\to H^2(X,\mathcal{O}_X)\to\cdots
\end{equation*}
But $H^2(X,\mathcal{O}_X)\cong H^{0,2}_{\bar{\partial}}(X)$ and since we are on a Riemann surface there no exist $(0,2)$-forms and therefore $H^{0,2}_{\bar{\partial}}(X)\cong 0$. In particular from this follows that $c_1$ is surjective and by (1),(2) above that any complex line bundle over a Riemann surface admits a holomorphic structure.
For your second question just note that $H^2(X,\mathbb{Z})\cong\mathbb{Z}$, so you can take your favorite integer $n$ and find a complex line bundle $L$ such that $\int_Xc_1(L)=n$. For example if $n>0$ take $\tilde{L}$ to be a degree $1$ line bundle and define $L:=\tilde{L}\underbrace{\otimes\dots\otimes}_{n \ \text{times}}\tilde{L}$ (note that $\mathbb{Z}$ is generated by $1$ and $-1$ as additive group)
One reference is Principles of Algebraic Geometry by Griffiths and Harris, namely part $2$ of the Proposition on page $141$. To see why that statement is equivalent to the one in your post, you will need to understand the divisor-line bundle correspondence which is explained in the preceding pages. In particular, see the first paragraph on page $136$.
Best Answer
Yes. This is known as the divisor - line bundle correspondence, see section 2.3 of Huybrechts' Complex Geometry: An Introduction for example.
A divisor in a complex manifold $Y$ is a locally finite, formal, integral linear combination of irreducible hypersurfaces of $Y$. That is, a divisor takes the form $\sum_Z \eta_ZZ$ where $\eta_Z \in \mathbb{Z}$, the sum is taken over all irreducible hypersurfaces $Z \subset Y$, and the collection $\{Z \mid \eta_Z \neq 0\}$ is locally finite. We say a divisor is effective if $\eta_Z \geq 0$ for all $Z$. An analytic subvariety of codimension one is therefore an example of an effective divisor.
On a complex manifold, one can construct a holomorphic line bundle $\mathcal{O}(D)$ associated to a divisor $D$, which admits a meromorphic section $\sigma$ whose associated divisor is $D$. If the divisor was effective, then the section $\sigma$ is actually holomorphic. In particular, the line bundle associated to an analytic subvariety $Z$ of codimension one admits a holomorphic section $\sigma$ such that $\sigma^{-1}(0) = Z$.
Beyond complex manifolds, you must be careful. There are two notions of divisors in algebraic geometry, namely Weil and Cartier; they do not coincide in general. The definition of divisor I gave above is actually the definition of a Weil divisor, while the notion of Cartier divisors is the one which gives rise to holomorphic line bundles with sections.