No. Any local system (vector bundle with constant coefficient transition matrices) admits a flat connection. Equivalently, local systems are given by representations of the fundamental group of the base. In your example, the Möbius bundle over $S^1$ admits a flat connection, since it arises from the sign representation of $\mathbb{Z}$ into $GL_1(\mathbb{R})$.

No. Any local system (vector bundle with constant coefficient transition matrices) admits a flat connection. You may simply use $d$ in each coordinate of a local trivialization, and the fact that the transitions have zero derivative makes this well-defined.

In fact, local systems are equivalent to representations of the fundamental group of the base. In your example, the Möbius bundle over $S^1$ admits a flat connection, since it arises from the sign representation of $\mathbb{Z}$ into $GL_1(\mathbb{R})$.