The Wigner function is the fourier transform in one variable of the density matrix for the single particle $\rho(x,y)$. If you Fourier transform in y, you get the Wigner function $\rho(x,p)$. This is important to understand, because it explains why the Wigner function is at all interesting, and why it obeys simple dynamics. It also shows that it doesn't have a probability interpretation away from the classical limit, because only the on-diagonal density matrix elements are probabilities.
Question 1: is is possible for Wigner's density to be negative after starting positive?
The answer is yes for a general potential, but for the special case of a Harmonic oscillator, the answer is no, because time evolution just rotates phase space. For the other special case of a free particle, a Gaussian wavefunction just spreads into a wider Gaussian, so it is not possible there either. For this reason, you have a hard time getting an example.
It is also true that semiclassically, the motion is along the classical trajectory, with shear according to the change in period with increasing J. So if a semiclassical Wigner matrix is positive near a single non-chaotic trajectory, it won't become negative, near the trajectory, at least not for a long time.
But it is very easy to get any kind of Wigner densities away from the semiclassical regime.
Question 2: What about the "probability density on phase space" interpretation?
This interpretation is faulty. The density matrix is only a probability for on-diagonal elements. So the Fourier transform of the Wigner density in either x or p has a diagonal probability density interpretation at all times, but the x,p position Wigner function is nothing. It's just a complicated encoding of off-diagonal matrix elements that is not particularly special or useful, away from the semi-classical limit.
To see that it is no good, consider a plane-wave state, whose phase space Wigner density is the plane-wave profile times a delta function at a certain value of p. If you scatter any lump into approximate plane waves, the end state of scattering is a bunch of complex stuff that has no interpretation as a phase-space probability.
The proper interpretation of Wigner's phase space density is that it is a Fourier transform of the density matrix, nothing more.