What is $\mathbb{Z}[x]/(x,x^2+1)$ isomorphic to?

Your reasoning is correct until the last line. $\mathbb{Z}[i]/(i) \cong \{0\}$, not $\mathbb{Z}$. Indeed, $(i)$ is the unit ideal in $\mathbb{Z}[i]$, since for any $a\in \mathbb{Z}[i]$, $a = (-ai)i$.

You could also note that $1=(x^2+1)-x(x)\in (x,x^2+1)$, so $(x,x^2+1)=\mathbb{Z}[x]$. From this point of view, the quotient is evidently 0.