Is there a known algebraic structure with two binary operations defined such that one of the operations behaves like multiplication [...] and the other behaves like an addition (identity and commutative) that is non-associative?
I'm sure there are many. One family of examples of commutative nonassociative algebras is given by Jordan algebras.
(Update: Sorry, I didn't realize you resolved the ambiguity to require that addition be nonassociative. Sorry, no ideas about that one. Never heard of that permutation.)
Generally speaking, is there an online index of algebraic structures searchable by features such as those indicated in the description above -or- is this site the best resource we have?
See this related question.
There is a (not very searchable) listing at http://math.chapman.edu/~jipsen/structures/doku.php/ of different types of structures.
You should probably also read the wiki article on algebraic structures. From time to time an table with a "binary operation" axis and a "property" axis with marks explaining what axioms are required appears. Usually they are quickly deleted since many people find them confusing, space consuming, and generally not useful.