I guess I’m confused by your notion of „diffusion“. For me the flux $j$ in the Cahn-Hilliard equation describes precisely the inter-diffusion of the two species.

The last sentence is only correct for ideal diffusion but not for the non-ideal mixtures described by the Cahn-Hilliard equation.

Doesn't the centrifugal force affect the foam the same way as the fluid? Your analysis didn't assume any mass or density, so the same argument should hold for the foam.

Interpreting entropy as disorder is dangerous, since there are cases where a liquid (the unordered phase) has lower entropy than the corresponding crystal (the ordered phase). Such systems crystalize when the temperature is increased!

Whether the equation is right depends on your definition of the surface tension. The equation given there assumes that the surface tension is an interfacial tension between soap and air. Conversely, one could define the surface tension as the one being measured when pulling on the soap film as argued by alarge. This would roughly lead to a surface tension twice as big, for which the above formula would have to be corrected.