The inclusion from a closed totally convex subset to the ambient manifold is a homotopy equivalence. Details can be found in "Totally convex sets in complete Riemannian manifolds" by Bangert, JDG, 1981.
For the second assertion, Cheeger-Gromoll prove in their paper on the soul theorem that any closed totally convex subset is a manifold with boundary, so if the boundary is non-empty, the manifold has zero top-dimensional homology, hence it cannot be homotopy equivalent to a closed manifold of that top dimension.
