This may sound lame, but I'd say you just look at the properties of the sequence you care about, and if you can define it so those properties still hold (exponent rules, recursion, universal properties...), then you do. At least I can't imagine there being a more general answer than this.

Regarding 0^0, I'd say 0^0=1 works better "algebraically", since then you can still write 0^0=0^(-0)=1/(0^0), and 0^0=0^(0+0)=(0^0)*(0^0).