While reading Demazure-Gabriel's construction of $\mathcal{S}ch$ as a full subcategory of $\mathcal{P}sh(CRing^{op})$, I've been trying to translate their exposition into the language of covering sieves and Grothendieck topologies. The requirement that a scheme be a sheaf in the Zariski topology on $CRing^{op}$ is essentially already in the language of sieves, but the requirement that there exists a cover of affines is giving me some trouble.

Question, then:

In general, is there a natural induced Grothendieck topology on $\mathcal{P}sh(\mathcal{C})$ or $\mathcal{S}h(\mathcal{C})$, where $\mathcal{C}$ is a site (not a pre-site!)?

Edit: I removed the other parts of the question regarding t-Schemes and algebraic spaces as functors of points to ask them at some other time.