There are an infinite number of different distribution functions that can be used to describe any situation of interest. You can have asymmetry, bumps, or even point-like delta functions in your distribution. In most realistic situations, collisions in the fluid will act to increase the entropy of the system by randomizing the particle velocities, and the normal distribution is the particular distribution that has maximum entropy. Thus, distributions in thermal equilibrium are isotropic normal distributions.
Away from equilibrium, a good way to quantify the asymmetry of the temperature is to define the pressure tensor:
$$
\overset{\text{$\leftrightarrow$}}{P}(x) = \int d^3v\; m\vec v \vec v f(x,v),
$$
which can carry a different temperature in each spatial direction, as well as off-diagonal elements that define the shear-stress. Often this quantity is split as
$$
\overset{\text{$\leftrightarrow$}}{P}(x) = p(x)\overset{\text{$\leftrightarrow$}}{I} + \overset{\text{$\leftrightarrow$}}{\pi}(x),
$$
where $p$ is taken to be the isotropic pressure and $\overset{\text{$\leftrightarrow$}}{\pi}$ is termed the stress tensor which measures the deviation of the distribution from spherical symmetry.
One can also define the heat-flux vector as the third-moment of the distribution,
$$
\vec q(x) = \int d^3v\; {m \over 2}\;(\vec v-\langle \vec v\rangle)^2\; \vec v \;f(x,v),
$$
which defines the flux-density of heat-energy.
To gain even further insight into the dynamics of this process, it is best to describe things in terms of kinetic theory where the distribution function is evolved in time according to the equation
$$
{\partial f \over \partial t} + \vec v \cdot \vec \nabla f + {\vec F \over m} \cdot {\partial f \over \partial \vec v} = \mathcal{C}[f].
$$
The collision term on the r.h.s. of the above equation is responsible for the randomization that leads to entropy production. This term forces the distribution towards a normal distribution on a time-scale on the order of the inter-particle collision-time. When one considers times much faster than this or situations where collisions can be neglected (i.e., a high-temperature plasma or a low-density fluid), the terms on the l.h.s. can dominate and the distribution can stray far from being normal.
We can also perturb the kinetic equation to determine the stability of a given distribution. If we perturb the given distribution and our perturbation damps out in time, we can conclude the distribution is stable. However, if our perturbation grows in time, then this implies the given distribution is unstable.
Including the self-consistent effects of electromagnetism/collisions complicates things further and are actively researched fields. These are probably out of the scope of this answer. But kinetic theory is definitely a worthwhile endeavor.
Hope this helps answer your questions.
