I have some questions of same flavour about two following constructions in Daniel Huybrechts's notes on K3 surfaces.

Construction 1: Kummer surface (Example 1.3 (iii), page 8) Let $k$ be a field of $char(k)\neq 2$ and let $A$ be an abelian surface over $k$. The natural involution $\iota: A \to A, x \to −x$, has the $16$ two-torsion points as fixed points. Let $\widetilde{A}$ be the blowup in these $16$ fixed points. It's easy to see that the involution $\iota$ lifts to $\widetilde{A}$. The quotient $\pi: \widetilde{A}\to X:=\widetilde{A}/ \iota$ is a ramified double covering of degree two.

Claim I: $\pi_* \mathcal{O}_{\widetilde{A}}= \mathcal{O}_X \oplus \mathcal{L}^{*}$
where $\mathcal{L}$ is square root of line bundle $\mathcal{O}(\sum_{i=1}^{16} E_i)$ - ie $\mathcal{L}^{\otimes 2}= \mathcal{O}(\sum_{i=1}^{16} E_i)$ - where $E_i$ are the exceptional divisors of the fixed points.

Construction 2: (Example 1.3 (iv), page 9) Consider a double covering $p: Y \to \mathbb{P}^2$ brached along a curve $C \subset \mathbb{P}^2$ of degree six.

Claim II: $ p_* \mathcal{O}_{Y}= \mathcal{O}_{\mathbb{P}^2} \oplus \mathcal{O}(-3)$.

Question: How to derive the two claimed formulas for the pushforwards of the structure sheaves? The second construction looks more "generic", in the sense one may pose more general question:

Assume more generally $f:X \to Y$ is branched covering of surfaces of degree $d$ brached in a curve $C$ of degree $s$. Is there a formula for pushforward sheaf $f_*\mathcal{O}_X$ on terms of $\mathcal{O}_Y$ and sheaves related somehow to branch locus $C$ known?

Motivation: There is a well known formula for dualizing sheaves for branched coverings. But I nowhere found a formula for pushforward of structure sheaf even if it looks more "natural" that such should exist.

What, if we generalize by dropping the surface assumption and replace $C$ by a smooth divisor.