Related to question 1), another example of some non-functional relations widely used in algebraic geometry is "rational functions" between varieties, side they are not defined everywhere, and in a sense may be thought as "multiply defined" (think of the birational map from a variety to its blow-up at some point).

Actually, traditional (e.g. italian) algebraic geometry put the notion of rational functions at the foundation of algebraic geometry. In a development that interestingly parallels the emergence of the concept of function as a fundamental notion in the nineteenth century, but one century later, the modern foundations of algebraic geometry (Weil, Zariski, and most importantly Grothendieck) moved the emphasis from the notion of rational functions to the notion of morphism.