Free surfaces occur when there are several phases present with sharp(ish) interfaces free to move. Some methods allow to treat all the phases as a single mathematical problem (e.g. level set or phase field), while some others track explicitly this free surface (e.g. ALE or immersed boundary methods).
Can it save computational time? If you use a method as ALE method and solve only for one phase while simplifying the model of the other (e.g. assuming a perfect fluid with uniform pressure), the computational domain will be smaller but will usually have a more complex time-evolving geometry. This may not save time. On the other hand, if quantities like density are very different between the two phases, the condition number of the linear system associated will be much lower with ALE-like methods.
Two parts here:
a) Is it more precise? This will all depend on the meshing, a refined mesh in the location of strong gradients of physical properties can also be efficient, although the thickness of "interfaces" in implicit interface methods generally needs to be a few times the mesh size there.
b) What about the boundary conditions at the interface? Complex boundary conditions are usually easier to write at an explicitly tracked boundary, because you can usually discretise them directly rather than rewrite them in terms of quantities such as the level set function.
Your formulation is misleading. If you have two homogeneous phases and solve only for one using ALE-like methods, yes you will go from space-dependent parameters in an implicit interface method to homogemeous-parameter equqtion, but in general you'll think of it the other way: tracking the interface leads to two problems in two sub-domains.
Explicitly tracking the interface will require you to model the (de)coalescence events rather than just "capture" them. In general, if notheing is done, (de)coalescence events will give rise to mesh self-intersection. You need to prevent this by an explicit numerical modelling in explictly tracked interface methods, while for implicit interface methods this will happen seamlessly -- which doesn't mean that this will happen with the right physics.