Check out the paper of Kontsevich-Rosenberg Noncommutative spaces and Noncommutative Grassmannian and related constructions. You will get what you want. i.e. the definition of properness and separatedness of presheaves(as functors, taken as "space") and morphism between presheaves(natural transformations).
Notice that these definitions are general treatment for algebraic geometry in functorial point of view,nothing to do with noncommutative.
Definition for separated morphism and separated presheaves
Let $X$ and $Y$ be presheaves of sets on a category $A$(in particular, $CRings^{op}$). We call a morphism $X\rightarrow Y$ separated if the canonical morphism
$X\rightarrow X\times _{Y}X$ is closed immersion We say a presheaf $X$ on $A$ is separated if the diagonal morphism:
$X\rightarrow X\times X$ is closed immersion
Definition for strict monomorphism and closed immersion
For a morphism $f$: $Y\rightarrow X$ of a category $A$, denote by $\Lambda _{f}$ the class of all pairs of morphisms
$u_{1}$,$u_{2}$:$X\Rightarrow V$ equalizing $f$, then $f$ is called a strict monomorphism if any morphism $g$: $Z\rightarrow X$ such that $\Lambda_{f}\subseteq \Lambda_{g}$ has a unique decomposition: $g=f\cdot g'$
Now we have come to the definition of closed immersion: Let $F,G$ be presheaves of sets on $A$. A morphism $F\rightarrow G$ a closed immersion if it is representable by a strict monomorphism.
Example
Let $A$ be the category $CAff/k$ of commutative affine schemes over $Spec(k)$, then strict monomorphisms are exactly closed immersion(classcial sense)of affine schemes. Let $X,Y$ be arbitrary schemes identified with the correspondence sheaves of sets on the category $CAff/k$. Then a morphism $X\rightarrow Y$ is a closed immersion iff it is a closed immersion in classical sense(Hartshorne or EGA)
Definition for proper morphism just follows the classical definition: i.e. universal closed and separated. You can also find the definition of universal closed morphism in functorial flavor in the paper I mentioned.
- 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.
- 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.
- 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.
Best Answer
You can try having a look at this paper:
http://arxiv.org/abs/math.AT/0011121
It's the most functorial-minded paper on algebraic geometry I'ver ever seen. It's written by an algebraic topologist. He cares mostly about affine and formal schemes.
The definition you're looking for is in section 4 of the paper. The functorial point of view for a formal scheme is a small filtered colimit of schemes, the colimit taken in the functor category.