Thank you so much for being so patient. This was very helpful.

Thanks a lot. If you don't mind me asking, I have tiny follow up questions. When you say "Stallings proves (1)" do you mean that it is apparent from his proof, or is it also stated explicitly in some Theorem/Lemma? When you say "Waldhausen proves (2)" do you mean that it follows from Theorem 6.1 of his article On irreducible 3-Manifolds which are sufficiently large (if not, what is the reference)? If yes, does his condition "boundary-irreducible" allow empty boundary or not (if only non-empty boundary is allowed, do you apply the theorem to $T_f \setminus {\rm Disc}$)?

2) If $T_f$ and $T_{f^\prime}$ are mapping tori of homeomorphisms $f,f^\prime:S \to S$ such that their fundamental groups are (abstractly) isomorphic, then $T_f$ and $T_{f^\prime}$ are diffeomorphic. I only care about the case where the given mapping tori are aspherical. This should follow immediately from Thurston's mapping tori theorem and Mostow rigidity. Is there a proof that does not use Thurston's theorem?

1) Let $p:M \to S^1$ be the bundle projection given by Theorem 2. Then the induced map $\pi_1(p):\pi_1(M) \to \pi_1(S^1)$ coincides (up to possibly precomposing with an automorphism of $\pi_1(M)$) with the given surjection $\pi_1(M) \to \mathbb{Z}$. I hope that this is an artefact of Stallings' proof. Is that correct?

Thank you for your answer. I'm not sure if I am obsessing over tiny details, but I have the impression that what is stated is not yet enough to conclude what I want. Theorem 2 gives me that $M$ is some mapping torus $T_{f^\prime}$. However, I would like to conclude that $M$ is $T_f$ with the exact $f$ I described in my question. But I guess this should be clear if one of the following two statements is true (which I state in the next comment):

I explicitely stated that I do not want to make use of Thurston's mapping torus result. But I will have a look at Waldhausen's theorem. Also, I would still be interested in the answer to my first question.