Perpendicular axis theorem is for flat bodies only. Never apply $I_z = I_x + I_y$ to a sphere or cylinder — it will give a wrong answer every time.

Rolling race winner is always the solid sphere. $K^2/R^2 = 2/5$ is the smallest; neither mass nor radius changes the outcome.

Disc about tangent in its plane: $5MR^2/4$. Two-step derivation: perpendicular axis gives $I_{diameter} = MR^2/4$, then parallel axis adds $MR^2$.

Contact point in rolling without slipping has zero velocity. The translational and rotational velocities at the contact point cancel exactly ($v_{cm} = \omega R$).

When a skater pulls arms in, L is conserved but KE increases. The extra kinetic energy comes from internal muscular work — this is not a violation of energy conservation.