Black hole, ingoing light-like and causality

Firstly, you can't globally define simultaneity or globally define a time coordinate $t$. It also appears like you're talking about the Schwarzschild $t$ and $r$ coordinates, which don't really make sense to use within the horizon $r_s$. We use the new timelike and spacelike coordinates $T$ and $R$, which are given explicitly on the Wikipedia page. These are defined on the whole spacetime and they keep their timelike and spacelike characters respectively.

You can also note that within $r<r_{s}$, in the Schwarzschild coords, the $t$ coordiante becomes spacelike and the $r$ coordinate becomes timelike, so the singularity at $r=0$ is actually a future singularity. Hence worldlines/trajectories within the horizon don't travel into the past (which is only defined locally anyway).