Let $M$ be a smooth $n$-manifold. Here are three constructions which produce manifolds which are homeomorphic to $M\times S^1$, but might not be diffeomorphic to it:
Take $M'\times S^1$, where $M'$ is homeomorphic to, but not diffeomorphic to $M$.
Take a connected sum of $M\times S^1$ with an exotic $n+1$-sphere.
Take the mapping torus of a self-diffeomorphism of $M$ which is homotopic, but not (differentiably) isotopic to the identity.
I would like to know two things:
Question A: when do these constructions actually produce new smooth structures?
Question B: which manifolds have the property that every smooth structure on them can be obtained by operations 1-3 above?
I would be interested both in theoretical answers with references to the literature, and explicit answers for specific manifolds, such as $S^6$, or other large-but-not-too-large dimensional spheres.
Remark 1: question A for construction 2 (connected sums) was essentially already asked here. This problem is apparently known among experts as "computing inertia groups". So I think part of my question is: are the inertia groups of $S^n\times S^1$ known?
Remark 2: question A for construction 3 is closely related to this question The difference is that I am asking about the diffeomorphism type of the total space, not fibered equivalence.