In continuation to my previous post: Question Regarding Riemann-Hurwitz Formula Proof

I'll be glad to receive some explanations regarding the following:

1. I know that when taking a sheaf $F$ , then the notation $f_* F $ corresponds to the direct image sheaf. In the answer Gunnar posted, he wrote: $f_* \mathbb{C} = \mathbb{C}^{ \oplus d} $ . What is the meaning of $f_* \mathbb{C}$ in this context? Does the notation $ \mathbb{C}^{ \oplus d}$ means the direct sum of d times $\mathbb{C}$ ?

2. Given the short exact sequence $ 0 \to f_* \mathbb{C} \to \mathbb{C}^{\oplus d} \to \mathcal{G} \to 0 $ .
   I understand from the previous calculation, the map between $f_*\mathbb{C}$ and $\mathbb{C}^{\oplus d} $ is the identity. But what is the map between $\mathbb{C}^{\oplus d} $ and the skyscraper sheaf $ \mathcal{G}$ ?

3. Given a Riemann Surface $X$ , does the algebraic Geometry of its genus is $\chi(X) := \chi(X,\mathbb{C})$ ?

Thanks in advance !