Given your examples, you don't seem to be asking for a canonical way to extend arbitrary functions defined on positive integers to zero. Instead, you're taking functions whose inputs are sets and asking if they can be defined when some input is the empty set. As long as your sequence defined on positive integers comes equipped with this extra structure, you shouldn't have too much trouble extending it naturally. If you start with an unstructured sequence, the reasons for favoring one extension over another become rather weak (e.g., Kolmogorov complexity).

Here's the standard example of a sequence that extends to zero in different ways: the sequence that is identically zero on the positive integers. One extension is the zero function. Other extensions interpret the sequence as n -> k 0ⁿ for some nonzero k.

Incidentally, you need to choose a base point on your space to define pi₀. Once you have that, it is the set of homotopy classes of pointed maps from S⁰ to your space. Equivalently, it is the (pointed) set of path components. It does not have a natural group structure (although it may if your space comes with some kind of composition law).