What is the name of a function $f$ such that $f(x,y) \in L \iff x\in L \wedge y \in L$?
They are typically called AND-functions. (I'm not joking.) Indeed, this concept has been considered before, and that's what people call them. See, for example, the book by Kobler, Schoning, and Toran on Graph Iso, where they talk about AND- and OR-functions for GI. And, by the way, there is an OR-function for GI (ibid.).

The question of an AND-function for graph automorphism is, I believe, still open :) (as stated in the book above).

Based on your last paragraph, the type of reduction you are talking about can also be generalized to what are called "truth-table" or "tt" reductions. These are non-adaptive Turing reductions (the queries are fixed by the input, but cannot depend on the answer to previous queries). For example, the negation kind of reduction in your last paragraph is a 1-tt reduction (1=number of queries).