Let's discretize time $t$. In other words, assume that we have $n$ Grassmann-odd variables $\theta^1, \ldots, \theta^n$. In this language OP's differential operator $\partial_t$ is replaced with some Grassmann-even matrix $A^i{}_j$. Define $$\theta^{\prime i}= \sum_{j=1}^nA^i{}_j \theta^j.$$ 

Then a straightforward calculation shows that OP's path-integral is replaced with
$$\begin{align} \int d\theta_n \ldots d\theta_1~\theta^{\prime 1}\ldots \theta^{\prime n}~=~&\ldots~=~
\int d\theta_n \ldots d\theta_1~\delta(\theta^{\prime 1})\ldots \delta(\theta^{\prime n})\cr
~=~&\ldots~=~\det(A),\end{align} $$
which indeed is a discrete version of OP's claim (2).