Subspaces of the Torus homeomorphic to $S^1$

Regarding your attempts at property 1, your explorations regarding $S^1 \times \{m\}$ look interesting, using the retraction $f : S^1 \times S^1 \to S^1 \times \{m\}$. But instead of postcomposing $f$ by a homeomorphism between $S^1 \times m \mapsto A$, you should have considered conjugating $f$ by a homeomorphism of $S^1 \times S^1$: if $g : S^1 \times S^1 \to S^1 \times S^1$ is any homeomorphism, then $g^{-1} \circ f \circ g$ is a retraction from $S^1 \times S^1$ onto the circle $g^{-1}(S^1 \times \{m\})$.

From that, perhaps you can leap to the following guess: if an embedded circle $A \subset S^1 \times S^1$ is a retract then $(S^1 \times S^1) - A$ is connected.

This gives a hint to a counterexample: look for a circle $A \subset S^1 \times S^1$ such that $(S^1 \times S^1) - A$ is disconnected, and prove that $A$ is not a retract.

Regarding property 2, here are a couple of things you may know: every deformation retraction is a homotopy equivalence; and every homotopy equivalence induces an isomorphism on the fundamental group. If you can compute the fundamental groups of the torus and of the circle, I think you'll be able to address property 2.