integrability of system of SDE relative to Langevin dynamics

Question

Similar to the system in this question and a follow up to here, consider the system (S) defined as \begin{align} {\rm d}X_t&=V_t\,{\rm d}t+\sigma_x\,{\rm d}W_t^1\\ {\rm d}V_t&=F_t\,{\rm d}t+\sigma_v\,{\rm d}W_t^2, \tag{S} \end{align} where $W_t^1$ and $W_t^2$ are two Wiener processes and $F_t$ is a gradient force from Hamiltonian $H$. If $\sigma_x=0$, then we have Langevin dynamics. On the other hand, if $\sigma_x \neq 0$, then what can we say about (S)?

Three questions:

1. Let $Z_t = (X_t, V_t)^{\top}$ Then is ${\rm d}Z_t$ an adapted process? So (S) is integrable?
2. Why or why can't we write (S) in a form of a second order ODE with perturbations like Langevin dynamics?
3. What does it mean for (S) to be non-integrable? Is there a simple way to make (S) non-integrable, and would we then use something like stochastic averaging methods to understand its behavior?

Thanks!