What are the 2-connected 6-manifolds that have boundary $S^5$? Are they all of the form $(\sharp_{i=1}^k S^3 \times S^3) \backslash D^6$ for some $k \ge 1$?
Also, I think if $M^5$ is simply-connected, has $w_2(M) = 0$, and is not $S^5$, then there is a unique 2-connected 6-manifold with boundary $M$. This seems to follow from Smale's classification of simply-connected 5-manifolds, Section 6.
http://www.jstor.org/stable/pdf/1970417.pdf?acceptTC=true.
But I have never heard such a claim before and it seems suspicious (especially in light of the $S^5$ case). Any references or counterexamples would be much appreciated.
Every diffeomorphism of $S^5$ extends across $B^6$ (because there are no exotic 6-spheres), so there is a unique way of capping off the boundary. Now you're just asking what the 2-connected 6-manifolds are. Smale classified these, and they are what you expected. See here.