In this definition, "scheme" means "affine scheme of finite type over $k$," where $k$ is a field. Product means fiber product over $k$.
$G$ is a group object in the category of $k$-schemes. A scheme $X$ with a $G$-action is a morphism of schemes (that is, of $k$-schemes) $G \times X \rightarrow X$ such that for all $k$-algebras $A$, if we identify $G \times X(A) = G(A) \times X(A)$ as sets, then
$$G(A) \times X(A) \rightarrow X(A)$$
is a group action.
First, what does it mean for $\pi: X \rightarrow Y$ to be "surjective on sets?" Does this mean that $\pi$ is surjective as a map of topological spaces? Or does this mean that for all $k$-algebras $A$, $\pi(A): X(A) \rightarrow Y(A)$ is surjective?
Second, if $f \in \mathcal O_X(\pi^{-1}(U))$, then $f$ induces a morphism of schemes $\pi^{-1}(U) \rightarrow \mathbb{A}^1 = \textrm{Spec } k[T]$, coming from the $k$-algebra homomorphism $k[T] \rightarrow \mathcal O_X(\pi^{-1}(U)), T \mapsto f$. Property (3) says there is a condition for $f$ to lie in $\mathcal O_Y(U)$. Does this mean that the map
$$\pi^{\#}(U): \mathcal O_Y(U) \rightarrow \mathcal O_X(\pi^{-1}(U))$$
is injective? Why is this?