Does anyone know if Fluent can solve free interface problem? And a more complicated interface is a thinless elastic membrane between two non-Newtonian fluids, can Fluent solve this complicated free interface? Thanks!

In your second question, does the membrane behave like the interface between a gas bubble and a liquid, or a liquid oil droplet in water, where the interfacial "skin" is held in place by the surface tension?

No, the membrane behaves like elastic thinless shell (the surface tension is changing with deformation, not constant as bubble liquid surface), the position is changing and deforming according to the around flow.

You may be able to represent the interface by the surface tension. Gradient adaption of the interface would minimise artificial diffusion. The geo-reconstruct interface tracking scheme with the VOF model would be your best bet.