Alternating series; first term is 0. Do I have a problem?

You can remove a finite number of terms and not affect convergence.

Observe that your series just rewrites $$ \sum_{n=1}^\infty (-1)^n \frac{n^2-1}{n^3+1}=\sum_{n=\color{red}{2}}^\infty (-1)^n \frac{n^2-1}{n^3+1}. $$

If your sequence is $a_n$, you could test the series for the sequence for $b_0 = 1$ (or $-1$) and $b_n = a_n$ for $n > 1$ for convergence.

You can "cleanly" apply the convergence test to $b_n$, and I will leave as an exercise relating that to the series for $a_n$ (it is not hard). Because this hack is so trivial we usually just apply it "sloppily," but you are definitely doing the right thing by asking how to do it properly.

In general when testing for convergence for any series you can do any arbitrary manipulation to the first $N$ terms you want (i.e. you "ignore" them, whatever that needs to mean).