Why does a corner slow a car down? Lets say we have a car, with power being given to maintain it at 100 km/h and about to negotiate a slight bend. As the car turns the corner, there is no change to the accelerator pedal, and no braking. All wheels maintain full traction. Let's assume it has completely stiff suspension (ie none) and the road is a perfectly flat surface.
Will this car slow down? If so, why? Racing cars still maintain a racing line (as straight as possible) through corners they can go flat out with full traction through, which is longer (but apparently faster) than if they hugged the inside corner. Why? What is the actual physical reason that causes cars to slow down in corners? Correct me if any of my assumptions are wrong.
 A: 
Will this car slow down?

In the ideal where the tires have sufficient grip, no it doesn't need to slow down.  In reality, the tires must slip a bit while turning, and this slip will introduce some energy loss that is not present in the straight-line driving.  But that could be minimal.

What is the actual physical reason that causes cars to slow down in corners?

It's because there is a maximum amount of force that the tire can apply to the ground before it begins to slip.  High-speed, tight-radius turns require much more force than high-speed, large-radius turns.
$$F_c = \frac{Mv^2}{R}$$
Increase the speed and decrease the radius, and the required forces go up a lot. 
If you have sufficient room, you'll probably try to keep your speed up and make a wide turn.  That way you don't have to accelerate as much after the turn.
If you don't have sufficient room, you'll try to lower your speed.  That reduces the required forces to something your tires can accomplish.
If you're not very good, you'll try to turn at too high of a speed.  When the required forces exceed the maximum grip of the tire, they slip and the car spins off the track instead of following the tight radius you selected.
