Distribution of dropped objects Consider small perfectly elastic spheres being dropped from a fixed height in R^3, bouncing and coming to rest on the horizontal R^2.  Assuming a reasonable distribution of minor perturbations of the initial velocity and minor perturbations of perfect elastic scattering when bouncing
can anything be said about the distribution  of final resting places ?  Make further simplifying assumptions without making it trivial.
 A: What you're asking about seems fairly close to that of what's called a ``freely-cooling granular gas'' which is of great interest in granular physics, geophysics, and even the study of the large scale structure of the universe.  I haven't found any literature on the specific situation you described of spheres falling under gravity, but since it seems you're interested in what happens when a lot of inter-sphere collisions occur, I think the physical phenomena described below should be quite relevant.  
Accordingly, in this answer I'll give a brief summary of some of the physics literature on this topic.
Freely-cooling granular gases are systems of hard sphere particles that interact via inelastic collisions, initially at a high "temperature" (particles with random velocities with some large average kinetic energy) and are studied as they dissipate energy and gradually come to a rest.  At short times the behavior is described by "Haff's law" where the particles remain homogeneously distributed and energy decreases in a power law fashion as $t^{-2}$.  
At longer times, the system enters a coarsening regime where it becomes inhomogeneous (clustering of particles) and the exponent of the energy decay changes to $t^{-\theta}$ where $\theta$ is dimension dependent.
A lot of attention has been given to dimension 1, where $\theta$ is known to be 2/3.  Ben-Naim, Chen, Doolen and Redner proposed that one-dimensional systems approach at long times the "sticky limit" (where the system behaves as if the coefficient of restitution is 0).  
The situation in 2 and 3 dimensions is still not clear although there have been many simulations and proposed theories.  Here is a recent paper by Pathak, Jabeen, Das and Rajesh with at least this nice image illustrating a simulation of the gas on the left, and another model called "ballistic aggregation" on the right.

Most of the literature regarding cluster sizes and such tends to be about the dynamics, i.e. how the distribution of clusters grows with time, etc.  
In fact, it is not obvious to me that the notion of "final resting places" is well-defined in these systems.
Inelastic collapse is an interesting phenomenon in systems of inelastically colliding particles, which is a singularity in the dynamics due to an infinite number of collisions occurring in finite time.  The existence of this also means that fining "reasonable distributions of minor perturbations" can be a very subtle affair.  Here's a paper by McNamara and Young describing simulations with this phenomenon in two and three dimensional systems.  They do have some discussion of how one might continue the dynamics through such singularities, but I haven't read it very carefully yet.
Note that the other papers above implement their simulations in a way so as to avoid these singularities altogether.  This is done by replacing inelastic collisions with elastic ones whenever the relative velocity between two particles is sufficiently small (see the paper of Ben-Naim et al).
