Should a polynomial satisfying certain conditions be linear?
No, try $h = t^2-t$ and $f = -4t^3 + 5t^2 + t - 1$.
If I have calculated this right, then $h$ is separable, $$f+f' = h \cdot (-4t-11),$$ $f$ and $f'$ do not have roots at $0$ or $1$ and so are coprime to $h$, and
$$\begin{align*} h'f &= 1 + h \cdot (-8t^2 + 6t + 3), \\ h'f' &= -1 + h \cdot (-24t+8). \end{align*}$$