Thank you! Your references answer my question, but I wanted to clarify something. The inertia group being trivial means that $W_g$ and $W_g \# \Sigma$ are only diffeomorphic if $\Sigma = S^n$ is standard. By Kreck's result that means and exotic $\Sigma$ must be different in $BO\langle k \rangle$-bordism from the standard sphere. So $\eta^B$ is injective and the kernel is zero. Agreed?

As I understand it Davydov-Yetter cohomology is about deformations of a given tensor category structure and not about existence, which is a very hard problem in general c.f. mathoverflow.net/a/285907/184.

The injectivity will fail even locally near any cusp point.

Even in the simplest connect sum case the classification of possible "framed knots" is a difficult and unsolved problem. When n=3 and k=1 this is a whole branch of topology. You certainly won't get a finite list of operations when $n \geq 3$.

@Joe No, this is not the case in higher dimensions. As Marco says, if the attaching sphere (aka the descending manifold of the elementary cobordisms' Morse critical point) is contained in a ball, then it determines a kind of "framed knot" in $\mathbb{R}^d$ or $S^d$. In that case the cobordism is equivalent to connect sum with the manifold which is obtained by surgery along this link in $S^d$. However if the attaching sphere is wrapped around the manifold non-trivially (e.g. maybe it represents a non-trivial class in $\pi_kM$) then you don't have a description in terms of connect sum.

Your description for k=0 as a connect sum with $S^1 \times S^{d-1}$ is only valid if the two red disks in the "before" manifold (= the "attaching manifold of the 1-handle") are in the same component. In the d=2 oriented case with k=1, the elementary bordism takes a surface M cuts out an embedded circle and glues on two disks. This operation either lowers the genus or, if the circle is "separating", cuts the surface in two. In particular it is not the operation of connect sum with another surface.

@Adrien You say "all relations formally follow from the genus 0 ones". Are you sure that is true? I don't think that is true once you are even in the (2,1)-category case. In particular in Cob(2) (as a (2,1)-category) I believe the punctured torus has automorphisms which cannot be recovered from the genus zero part. This would be the "type III" maps in the Hatcher-Thurston presentation of MCGs.

@DavidESpeyer I don't believe any such example exists. If there were such an X with non-trivial obstruction, then this obstruction would persist on the 3-skeleton of X. So we might as well assume X is a 3-dimensional CW complex. But then by cellular approximation any map from X to BU(2) factors through the 3-skeleton of BU(2), which happens to also be the 2-skeleton $\mathbb{CP}^1$. However the universal bundle already splits into two lines when restricted to $\mathbb{CP}^1$.

This is a good example. I was thinking about oriented manifolds and should have specified that in the OP. So this is one more example of what I was looking for. Is this plus what I listed the complete list of known examples?