the answer to your first question is of course no. For an counter example, it is enough to consider ordinary closed 1-forms, and you see that a necessary condition for the positive answer would be that all periods of both forms are the same. In the higher rank case, periods are replaced by the monodromy..

The answer to your first question is of course no. For a counter example, it is enough to consider ordinary closed 1-forms, and you see that a necessary condition for the positive answer would be that all periods of both forms are the same. In the higher rank case, periods are replaced by the monodromy.

More concretely (after Anton Petrunin's comment): Consider the case of $S^1=\mathbb R/2\pi\mathbb Z$ and the trivial line bundle over it with connections $\nabla_0=d$ and $\nabla_1=d+d\varphi.$ Consider also a connection $\tilde\nabla=d+\omega$ on $\mathbb C\to S^1\times [0;1]=:\tilde M$ which restricts to $\nabla_0$ respectively $\nabla_1.$ Then, the curvature of $\tilde\nabla$ is $d\omega$, and by Stokes theorem we get $$2\pi=\int_{\partial\tilde M}\omega=\int_M d\omega.$$ Hence, the curvature of $\tilde\nabla$ does not vanish identically.