Homology of complement of a $m$-sphere in $\mathbb R^n$, with $m<n$

I haven't read the linked question, so I won't suggest alternative approaches, but:

• Yes, you are using Mayer-Vietoris correctly/usefully. Of course when $k$ is very small you might have to deal with some nontrivial maps, but such is life.

• I believe the homotopy equivalence intuitively works like this. For a visual take $n=4, k=2$. Note $S^{n-1}-\{*\}$ is $\Bbb{R}^{n-1}$. Thicken the $(k-1)$-sphere to a $k$-ball with the origin removed, cut that out of the space (so the origin remains). Homotope the whole thing down to the unit $(n-1)$-ball with a hole in a coordinate plane, the shape of a "$k$-dimensional diameter", except the origin. Finally, expand the hole in the $(n-1)-k$ orthogonal dimensions until it meets the boundary $B^{n-1}$. Doing all this, you are left with an $(n-2)$-sphere that has a linear subspace [segment] running through it, and this is homotopic to $S^{n-k-1}\vee S^{n-2}$ by pulling the subspace segment outside and then homotoping the attaching sphere to a point.

• (A special argument may be needed for $n=2$ to deal with non-connectedness; also $n=1$ might be false as stated)