My own thought tend to revolve around some subset of the following:
--Find a combinatorial definition for the sequence, and see if it makes sense when you extend slightly further.
--If you are trying to perform a vacuous task (e.g. tiling an empty board, or counting functions defined on the empty set), you can do it in exactly one way. Most of your examples fall under this class, including a^0 (functions defined on the empty set), 0! (bijections on the empty set), F_1 (tiling an empty board), and the cardinality of the direct product of no groups (choosing one object from each class, so the direct product should be the identity).
--An empty sum is equal to 0, an empty product is equal to 1. (again the cardinality of the direct product of 0 groups should be 1).
What about the determinant of a 0x0 matrix? Well, it's a sum over all permutations from a 0 element to itself of an empty product. There's one element in the sum (vacuous task), and its an empty product, so should be 1.
I don't really know if there's a rigorous statement of this, or if there's not some way it can come into self-contradiction if there's two combinatorial ways of defining a sequence, but it's what seems natural to go by.