Negative simple path NP-Complete
You can reduce from the Hamiltonian path problem. Given an instance of the Hamiltonian path problem with $n$ vertices:


Add a start vertex $s$ and an ending vertex $t$.
For every vertex $v$ that is not $s$ or $t$, add edges $(s, v)$ and $(v, t)$ with weight $n-1$ and $n-2$ respectively.
For every other edge, assign it weight $-2$.


Now the previous graph has an Hamiltonian path if and only if there is a simple path with negative total weight from $s$ to $t$ in the new graph.