I'll answer your questions in reverse order:
There is no difference in the fluid dynamics of this problem (may not always be true as boundary layers may be different, but that's not a factor really here) if the car is moving forward at speed $V$ into quiescent air or the car is stationary with a wind moving towards it at speed $V$. It's just a change of reference frame which in this case isn't an issue.
Now for the actual hard part...
Your car is not going fast enough to make the flow compressible so we can assume it isn't. We can also assume that the flow doesn't have heat added/removed and for the most part inviscid. This all means that the flow is adiabatic and reversible, aka isentropic.
The isentropic assumption is a handy one because it allows us to say that total pressure is constant along streamlines. Since we'll assume the atmosphere is uniform (which for this works well enough, it is pretty uniform over the width of the car) this means that the total pressure is the same everywhere upstream where the flow isn't moving. Which means the total pressure is constant in the entire flow when moving.
Now, because it's incompressible and isentropic, we can define the total pressure as:
$$P_0 = P_s + \frac{1}{2}\rho V^2$$
which is Bernoulli's Principle where $P_0$ is the total pressure, $P_s$ is the static pressure, $\rho$ is the air density and $V$ is the air (or car) speed. Since the left hand side is constant, this tells us that as we go faster the $\frac{1}{2}\rho V^2$ term gets bigger which means $P_s$ gets smaller.
The air inside your car is, for our purposes, not moving. Or if it is moving, it's moving far slower than the air outside the car and can be assumed still. This means the static pressure inside your car is equal to the total pressure (at least initially, the pressure inside will decrease with the window open as the flow begins to move). However, because outside the air is moving with speed $V$, we know that the static pressure outside the car is lower.
This difference in the static pressures from inside the car to outside creates flow from inside the car to outside. So the flow brings with it the smoke and that's why it leaves the window.