POTW #02: The PH-SL0 Droid

$\textbf{Note:}$ All parts can be solved using standard techniques from the high school physics Olympiad syllabus. $\textbf{Optional Story:}$ PhysElo is designing a new grading droid with a spherical base and a conical computing top. Unfortunately, whenever it reads a terrible physics proof, the droid tips and wobbles uncontrollably across the desk. Please help the team must analyze its stability and oscillatory mechanics before deploying the unit: $\textbf{Problem:}$ A solid sphere of radius $R$ is truncated horizontally at a distance $z = R/2$ above its geometric center. A solid right circular cone of height $H$ is attached to the sphere such that its base coincides with the flat cut. The combined object has uniform density $\rho$, total mass $M$, and rests upright on a horizontal floor.