I was reading Thurston's "A norm for the homology of 3-manifolds" and I had some questions that I think are pretty basic but have me stumped at the moment. Let $M^3$ be a compact 3-manifold that fibers over $S^1$ and let $\Sigma$ be the fiber. Let $\Sigma'$ be an incompressible surface in $M^3$ that is in the same homology class as the fiber $\Sigma$.
After introducing his norm, Thurston goes on to prove that $\Sigma'$ is isotopic to $\Sigma$ in section 3 of the above paper. At the beginning of this section, Thurston gives a brief argument for why $\Sigma'$ must be homotopic to $\Sigma$. This homotopic part is the part that I do not understand and I wanted to ask about.
The first thing Thurston asks us to do is to look at the covering induced by projection to $S^1$ which is homeomorphic to $\Sigma \times \mathbb{R}$. Apparently, $\Sigma'$ lifts to this cover (this is presumably because of the condition that it is homologous to the fiber but I do not understand why this is) and then composing this lift with the projection $\Sigma \times \mathbb{R} \to \Sigma$, we have a map $\Sigma' \to \Sigma$.
Now apparently this map is degree 1, although it is not clear to me why. Then Thurston says that since $\Sigma \to \Sigma'$ is a degree 1 map that is $\pi_1$-injective, it is homotopic to a homeomorphism (which is intuitively believable although I would not know how to prove it). Then lifting that homotopy to $\Sigma \times \mathbb{R}$ and projecting it to $M$ gives the desired result.
Just to summarize my questions:
(1) Why does $\Sigma'$ lift?
(2) Why is the lift composed with the projection degree 1?
(3) Why is the degree 1, $\pi_1$-injective map $\Sigma' \to \Sigma$ homotopic to a homeomorphism?