Yes, even though some answers to nearly equivalent cousins of this question claim that the answer is No.
We really understand the right "theory of nearly everything" which means that using the microscopic theory – it's the "statistical mechanics" part that is most important here – we may calculate everything that happens in the world of everyday phenomena.
The general proof based on statistical mechanics that the entropy is increasing with time is known as Boltzmann's H-theorem
https://en.wikipedia.org/wiki/H-theorem
and there exist many versions and adjustments of this theorem and its proof – for example, in quantum field theory.
The fact that one may prove that entropy increases with time looks counterintuitive to many people because the underlying microscopic laws are time-reversal-symmetric (or at least, CPT-symmetric which makes the result pretty much equally counterintuitive). The flawed reasoning that suggests that time-reversal-breaking laws cannot emerge from time-symmetric microscopic laws was originally known as Loschmidt's paradox.
Like always in physics, there is no paradox. The reason why totally time-reversal-asymmetric – irreversible – phenomena emerge from a fundamentally T-symmetric theory is that the right laws of probability (in statistical mechanics) are inherently T-breaking even if the dynamical laws preserve T. The probabilistic reasoning breaks the symmetry, it respects the logical arrow of time (or psychological arrow of time, different terms are used).
The true mathematical origin of the asymmetry is that when we compute the probability that the ensemble of microstates A evolves into the ensemble B, we are summing over the microstates in the final state B (because the probability of "B1 or B2" is the sum of the probabilities for B1 and B2 separately) but we are averaging, perhaps with some weights (priors) over the initial microstates A1, A2. We need to average and not sum because the total probability of all initial microstates has to be kept fixed, at 100 percent, so they have to share the probability.
For this reason, we may rewrite the probability as the sum over all microstates in A as well as B as long as the result is divided by the number $N$ of microstates in A. But it is not divided by the number of microstates in B. This asymmetry encourages the evolution from lower-entropy initial states to higher-entropy final states. It is easy to see that the evolution of the T-reverted (or CPT-reverted) process is $\exp((S_B-S_A)/k)$ times less likely because the $\exp(S/k)$ factors arise from the $N$ I mentioned previously. If $S_B-S_A$ is macroscopic (of order one or higher in SI units), the process with the decreasing entropy is infinitely less likely than the increasing-entropy counterpart which means that it is impossible in the $k\to 0$ limit.
So yes, although the calculations are difficult and sometimes only doable in principle, we may not only prove that the entropy goes up but also to calculate how quickly the heat spreads from the warmer object to the cooler one, how quickly diffusion takes place in a specific dirt pumped into a specific liquid – how quickly any process involving the increase of entropy occurs. All these data are completely dictated by the microscopic theory. They have to be – otherwise the microscopic theory would be heavily incomplete, of course.
The reason why the entropy goes up has nothing whatever to do with cosmology or some assumptions about the Big Bang. The increasing entropy is a derivable result for any system, whether it's the whole Universe observed for billions of years or just its small part observed for picoseconds.
Your particular problem is easily solvable. The freely falling object behaves indistinguishably from an object outside a gravitational field. So if its internal degrees of freedom were in equilibrium at the beginning, they will stay in equilibrium and the entropy doesn't increase at all.
