Boy, I like this line of inquiry.  I've been thinking about similar issues myself for quite a while now, but I did not realize that so much work had already been done.  I'm not sure what you're looking for exactly, but I believe that categorical definitions for open and proper might be found in either Z. Luo's inspirational page www.geometry.net/cg  or in Dier's book Caegories of Commutative Algebras upon which Luo's work is based.  Both give answers to your question in the case of affine schemes.  I'm not sure if the answers translate to the full category of schemes, which I think is what you're looking for?