Actually, for symmetric $\mu$ (i.e. $w=$mean width), $\sqrt[3]{vol}/w$ is (globally) maximized by balls (aka Urysohn's inequality). For the standard width, the ball is locally neither maximal (see e.g. ellipsoids) nor minimal (e.g. bodies of constant width). What makes my $\mu$ different is the condition that the projection to the space of spherical harmonics of degree $n$ vanishes only for $n=1$. This is certainly untrue for the mean width (vanishes for all $n>0$) or for the standard width (vanishes for odd $n$). If it helps, my $\mu$ is supported at $12$ points with equal weight at each.

It is true for the minimum width, of course. Can you explain what you mean by "generalized widths" and also by "completely symmetric $\mu$"? The minimum width corresponds to the above $f_2$ with a measure supported at a pair of antipodal points.

For starters, it cannot have any complex roots.