What does $\sigma(t_0, \ldots, t_{n-1}) \in \mathcal{T}_{\Sigma}^{0}$ in this paper mean?
You're over-thinking it. It just says that, if $t_0$, ..., $t_{n-1}$ are in $T^0_\Sigma$ and $\sigma$ is a symbol in the alphabet $\Sigma_n$, then the string consisting of $\sigma$, followed by $($, followed by... is in $T^0_\Sigma$