Quantum mechanics, as well as quantum field theory or classical mechanics, are nothing but frameworks that are used to predict results given an experiment; as such, the question should not be whether or not some framework hold in some cases, but rather if it gives useful and correct predictions.
In classical physics, where classical means small velocities and small energies, one can actually measure classical observables (position and momentum, energy, temperature and the like) without affecting them very much (namely without having the problem that measuring position will increase the uncertainty on momentum): as such, one asks questions like where the particle is, what the velocity is, what is the total angular momentum.
Is it true that put any particle (non-charged; no spin; only translational and potential energy considered) on a force field $U$, its probability distribution can be calculated through Schroedinger Equation?
Yes, you can always calculate things given some initial frameworks, the actual question is what you can do with it: probability distributions are used to calculate scattering amplitudes in high energy experiments where (for some reasons) one cannot calculate the trajectory of the particles but only derive some indirect measurements (decay rates and similar). If the particle is "classical", you had better just look at the actual position and state where the particle is.
I am curious because if the answer is yes, some easy computer simulation will give interest quantum effects.
How does this have anything to do with quantum mechanics?