The problem with this question is that static friction and kinetic friction are not fundamental forces in any way-- they're purely phenomenological names used to explain observed behavior. "Static friction" is a term we use to describe the observed fact that it usually takes more force to set an object into motion than it takes to keep it moving once you've got it started.
So, with that in mind, ask yourself how you could measure the relative sizes of static and kinetic friction. If the coefficient of static friction is greater than the coefficient of kinetic friction, this is an easy thing to do: once you overcome the static friction, the frictional force drops. So, you pull on an object with a force sensor, and measure the maximum force required before it gets moving, then once it's in motion, the frictional force decreases, and you measure how much force you need to apply to maintain a constant velocity.
What would it mean to have kinetic friction be greater than static friction? Well, it would mean that the force required to keep an object in motion would be greater than the force required to start it in motion. Which would require the force to go up at the instant the object started moving. But that doesn't make any sense, experimentally-- what you would see in that case is just that the force would increase up to the level required to keep the object in motion, as if the coefficients of static and kinetic friction were exactly equal.
So, common sense tells us that the coefficient of static friction can never be less than the coefficient of kinetic friction. Having greater kinetic than static friction just doesn't make any sense in terms of the phenomena being described.
(As an aside, the static/kinetic coefficient model is actually pretty lousy. It works as a way to set up problems forcing students to deal with the vector nature of forces, and allows some simple qualitative explanations of observed phenomena, but if you have ever tried to devise a lab doing quantitative measurements of friction, it's a mess.)
I can´t fully come up with an explanation from more basic principles, but in the case you describe you will have kinetic friction. Or at least that is what all engineering books say...
There are a number of situations where this effect is clearly demonstrated:
- Pulling a cork out of a bottle, using a basic corkscrew such as this: if you simply pull on the corkscrew, it is harder to pull it out than if you first get it rotating and then pull.
While not entirely the same, a similar situation arises when a car rolling down a road with a strong side wind brakes and locks the wheels. While the wheels were rolling, there is no relative motion between the road and the tire, so there is static friction in effect, and unless the wind is really strong, as in a hurricane, the force will not overcome friction and the car will not skid sideways. Once the brakes are locked, the car starts skidding forward, because the force due to the inertia of the car overcomes friction. Once this happens, the car will also start skidding sideways, due to two factors:
the lesser important is that the coefficient of friction in effect once there is relative motion is the kinetic, not the static one.
the main effect is due to the fact that the direction of movement doesn't really matter at all: at the contact point you have a force due to inertia and a force due to the side wind, and once their combined magnitude exceeds friction, you will start having movement in the direction of that combined force, so forward but also to the side.
Best Answer
You are mixing up concepts.
The kinetic friction of a car has nothing to do with $F_N$.
The concept of friction as taught in school tells you, that friction is independent of velocity. This is clearly not realistic for a car! The friction - let's rather call it "air resistance", for this is the main part - has a very complicated dependence on the velocity - but it usually rises if you drive faster :)
You can keep at constant speed if all forces on you balance: the air resistance, that tries to stop you and the force that pushes you forward.
What is this force? Again friction. If there were no friction, the tires would move backwards on the road. So this is the force pushing you. For this friction the concept of static friction is suited in a good approximation.
So to you side-question: yes, for the concept of friction you learned in school, it will remain the same if you fix $F_N$ and $\mu$ - in particular independent of velociry. But this is not how reality works.