1. The highbrow way of reformulating your question is as follows. Consider the category $Sch$ of all schemes endowed with the Zariski topology. There is a fully faithful embedding of the category of affine schemes $Aff = CommRing^{op}$ into $Sch$; the topology induced on $Aff$ by that on $Sch$ is also the Zariski topology. The comparison lemma ([SGA4] III, 4.1) then says that, because any object in $Sch$ can be covered by objects in $Aff$, the category of sheaves on both sites are equivalent. In particular, representable sheaves in $Sch$ (i.e., schemes) are determined by their values on affine schemes.
2. For a sheaf $F$ on $Aff$ to be represented by a scheme it is enough that it be covered by affine schemes, i.e., that there exists affine schemes $U_i$ together with open immersions $U_i \to F$ (you have to define what this means, of course) such that $\coprod_i h_{U_i} \to F$ is an epimorphism of sheaves. Actually, you can take this a definition of scheme. The compatibility of the gluings in the classical definition is taken care of here by the sheaf condition.
3. Algebraic spaces can be similarly defined. While I was writing this, Harry beat me to giving the reference to the excellent notes of Bertrand Toën from a course of his on algebraic stacks.

In 2, you also ask if you can construct schemes from $Aff$ without actually using the fact that you are dealing with commutative rings. I think not. The categorical nonsense can get you only so far: at some point you have to introduce the geometry itself, and that is given by the $Aff$ with its topology. If you replace $Aff$ by the category of open sets in some $\mathbb{R}^n$ with open immersions you would end up defining manifolds. This is what Toën calls geometric contexts.

1. The highbrow way of reformulating your question is as follows. Consider the category $Sch$ of all schemes endowed with the Zariski topology. There is a fully faithful embedding of the category of affine schemes $Aff = CommRing^{op}$ into $Sch$; the topology induced on $Aff$ by that on $Sch$ is also the Zariski topology. The comparison lemma ([SGA4] III, 4.1) then says that, because any object in $Sch$ can be covered by objects in $Aff$, the categories of sheaves on both sites are equivalent. In particular, representable sheaves in $Sch$ (i.e., schemes) are determined by their values on affine schemes.
2. For a sheaf $F$ on $Aff$ to be represented by a scheme it is enough that it be covered by affine schemes, i.e., that there exist affine schemes $U_i$ together with open immersions $U_i \to F$ (you have to define what this means, of course) such that $\coprod_i h_{U_i} \to F$ is an epimorphism of sheaves. Actually, you can take this as a definition of schemes. The compatibility of the gluings in the classical definition is taken care of here by the sheaf condition.
3. Algebraic spaces can be similarly defined. While I was writing this, Harry beat me to giving the reference to the excellent notes of Bertrand Toën from a course of his on algebraic stacks.

In 2, you also ask if you can construct schemes from $Aff$ without actually using the fact that you are dealing with commutative rings. I think not. The categorical nonsense can get you only so far: at some point you have to introduce the geometry itself, and that is given by the $Aff$ with its topology. If you replace $Aff$ by the category of open sets in some $\mathbb{R}^n$ with open immersions you would end up defining manifolds. This is what Toën calls geometric contexts.