ag.algebraic geometry - Induced Grothendieck topology on a presheaf or sheaf category of a site?

The answer to your first question is yes. Suppose $C$ a site.

The category $C^{sim}$ of sheaves on $C$ simply has the canonical topology (SGA 4, Vol. 1, Exp. II, 2.5). The sheaves here are precisely the representable sheaves. (I'm ignoring questions about universes.)

The category $C^{wedge}$ of presheaves on $C$ inherits a topology in the following manner (SGA 4, Vol. 1, Exp. II, §5). Declare a morphism $Fto G$ of presheaves on $C$ to be a covering morphism if the induced morphism $aFto aG$ of associated sheaves is an epimorphism. Now say that a collection of morphisms $F_ito G$ is a covering family if the morphism $amalg F_ito G$ is a covering morphism. This gives $C^{wedge}$ the finest subcanonical topology such that covering families in $C$ give rise to covering families in $C^{wedge}$. This topology on $C^{wedge}$ can be seen as the lift of the topology on $C^{sim}$ described above along the sheafification functor $a$.