Honestly, not that many people in math think about the construction of the natural numbers frequently. Yeah, we learn about it of course, but that's about it. Very often the natural numbers do not include 0. Often they do. I've seen $\mathbb Z_{\geq0}$ and $\mathbb Z_+$ used to avoid having to worry about it.
Hell, different countries can't even agree on whether 0 is positive. In France, 0 is considered both positive and negative. In USA, 0 is considered neither.
(quick edit: I realize my last paragraph makes the $\mathbb Z_+$ option seem weird. I do math in the States.)
Only if you need zero, which you don't need for indexing. Peano's original axioms started from 1 (which is logically equivalent to starting from 0, or -1, or potato) if all you are doing is counting)
