Does minimal degree $n$ imply a $K_n$ minor

More generally, it is a classic result (independently due to Kostochka and Thomason) that minimum degree $(\alpha+o(1))n \sqrt{\log n}$ suffices to force a $K_n$ minor, where $\alpha$ is an explicit constant. Conversely, there are random graphs with minimum degree $\Omega(n\sqrt{\log n})$ that do not contain a $K_n$ minor. See here to access the paper by Thomason.

No.

The edge-graph of the icosahedron is regular of degree five, but does not have a $K_5$ minor because it is planar (Kuratowski's theorem).