Finite morphism of varieties - morphism of sheaves

Yes, the map is injective assuming the map is dominanat. Since $1\in\mathcal{O}_Y$ goes to $1\in f_*\mathcal{O}_X$, you should be able to check this.

The exact sequence above may not split in positive characteristic, but it does in zero characteristic (or characteristic not dividing $d$), using the trace map.

For the last part (char $\neq 2$), one has $f_*\mathcal{O}_X=\mathcal{O}_Y\oplus L$ and checking the algebra structure on $f_*\mathcal{O}_X$, one can show that $L^2=\mathcal{O}_Y(-E)$ where $E$ is the branch locus.

For the first part of your question, a morphism of schemes $f : X \to Y$ such that $\mathcal{O}_Y \to f_*\mathcal{O}_X$ is injective in said to be schematically dominant. If $f$ is dominant and $Y$ is reduced then $f$ is schematically dominant (EGA IV3 prop 11.10.4).