Let $X$ be a smooth projective curve over a field $k$ of characteristic zero. The algebraic de Rham cohomology of $X$ is, by definition, the hypercohomology of the complex of Kähler differentials for the Zariski topology:

$H^k_{DR}(X)=\mathbb{H}(X, \mathcal{O}_X \to \Omega^1_X \to \Omega^2_X \to \cdots)$

In "Hodge cycles on abelian varieties", p. 24, Deligne claims

"For a complete smooth curve $X$ and an open affine subset, the map

$H^1_{DR}(X) \to \Gamma(U, \Omega^1_X) / d\Gamma(U, \mathcal{O}_X)$

is injective with image the set of classes represented by forms whose residues are all zero (such forms are said to be of the second kind)."

I have several questions regarding this quote:

1) How does one prove the statement?

2) How to use this to determine $H^1_{DR}(X)$?

3) I'm a bit confused with the terminology "second kind". I thought this was reserved for $H^1(X, \mathcal{O}_X)$ whereas "first kind" are differentials in $H^0(X, \Omega^1_X)$. What does it mean?

4) related to 3) how does one see the Hodge decomposition in this setting?

Any help would be appreciated. Thanks a lot!