I had fun trying to make this as intuitive as possible. I hope I've succeeded without doing the physics of the situation much injustice.
When a car is driving straight ahead, the plane in which the wheels are rotating is aligned with the direction of movement. Another way of saying this is that the rotation axis is perpendicular to the momentum vector $\vec{p}=m\vec{v}$ of the car. So the friction merely makes it harder for the car to move, which is part of the reason why you need to put your foot on the gas pedal to maintain a constant speed. At the same time, the friction is what allows you to maintain that constant speed because the rotating tires sort of grab onto the ground, which is the intuitive picture of friction. The tires grab the ground and pull/push it backwards beneath themselves, as you would do when dragging yourself over the floor (if it had handles to grab onto). Those grabbing and pulling/pushing forces are what keeps you going.
Things change when the wheels are turned. The plane in which they are rotating now is at an angle with the direction of motion. Alternatively but equivalently, we could say the rotation axis now makes an angle with the momentum vector of the car. To see how friction then makes the car turn, think again in terms of the wheels grabbing onto the ground. The fact that they now make an angle with the direction of motion, means the force the tires are exerting is also at an angle with the direction of motion - or equivalently, the momentum vector.
Now, a force is a change in momentum$^1$ and so (because the wheels are part of the rigid body that is a car) this force will change the direction of the car's momentum vector until it is aligned with the exerted force. Imagine dragging yourself forward on a straight line of handles on the floor and then suddenly grabbing hold of a handle slightly to one side instead of the one straight ahead. You'll steer yourself away from the original direction in which you were headed.
$^1$ Mathematically: $$\vec{F}=\frac{d\vec{p}}{dt}$$
The wikipedia article on rolling resistance points out that there are several mechanisms at work. While the scenario illustrated in your diagram (deformable tyre on a hard road) can be modelled as a torque opposing the rotation of the wheel (also friction at the axle), other scenarios (eg hard wheel on a deformable road) might not be.
The distinction between friction force, normal reaction and rolling resistance is an artificial one, not inherent within nature, so it is not sensible to be pedantic about the differences between them, unless the distinction is made in the question.
Like friction, rolling resistance is related by some empirical law to parameters such as the normal reaction between the surfaces, the diameter of the wheel, tyre pressure, or the linear velocity of the vehicle. The law may contain several coefficients, and relates to a specific combination of materials. The wikipedia article provides examples.
Unless there is some particular reason to do otherwise, I suggest that the simplest solution is to model all forms of rolling resistance (whatever the mechanism) as a single force which opposes rolling motion.
In answer to your final question, there could be different combinations of rolling resistance and static/kinetic friction on each tyre of a moving vehicle, depending on the circumstances.
In the ideal case (no rolling resistance), constant velocity requires no force, so there is no static friction. If rolling resistance is not zero then there must be a driving force and therefore some static friction acting on all wheels to keep them moving at constant speed. If the vehicle is accelerating/decelerating then static friction is required to speed them up. If the vehicle is braking there may be some kinetic (sliding) friction on the braking wheels.
If all the wheels are the same, and bear the same load, then the rolling resistance is assumed to be the same on each, whether they are driving or braking or neither. Some wheels may bear a greater load, eg because the heavy engine is closer to them, or the car is accelerating or braking heavily. For those wheels rolling resistance will be higher.
Best Answer
When a car is braking, the wheels are pushing on the asphalt in opposite direction then when accelerating. So it is not the same story.
Acceleration: the force is generated by the engine which is applying torque on the shaft, which is turning the wheels. The friction at the contact point between the wheels and the road gives rise to the forward force.
Braking: during braking, the engine is usually not actively applying torque to the shaft, so the forward force is gone. On the other hand, the brake pads are applying force on the rotor, slowing it down. This has the opposite effect on the wheel rotation: they are not forced to turn faster, but slower. This (+ friction) gives rise to decelerating force at the contact point with the road.