Are there two forces applied by the road to an accelerating car? A car is travelling straight in one direction. If the car is accelerating (a positive one, so it becomes faster) on a nonfrictionless road, then the ground exerts two forces to the car, each in opposite direction, to the car, right? The first one is required to accelerate the car, and the second one is the static friction. The two forces are opposite in direction, but different in magnitute. The first force is greater (in magnitude) than the second one (static frictional force) in order for $\sum F_x >0$ and thus it's accelerating.
 A: The car accelerates because of the friction alone. The only reason the road can exert a force on the car to accelerate it is because the car is exerting a force on the road to accelerate it in the opposite direction.
If there was no friction between the tires and the road, the car would stand still.
A: 
then the ground exerts two forces to the car, each in opposite
direction, to the car, right?

No. The ground only exerts one force, and that's the static friction force acting forward on the wheel. The other force is exerted by the wheel on the ground acting backward. The two are equal and opposite per Newton's third law.

The first one is required to accelerate the car, and the second one is
the static friction.

The force that accelerates the car is the static friction force. It is the only external force acting forward on the car and is therefore responsible for its acceleration per Newton's second law. That force is the equal and opposite reaction to the force the wheel exerts backward on the ground per Newton's third law. The force the wheel exerts on the ground is responsible for accelerating the earth, per Newton's second law, as discussed below.

The first force is greater (in magnitude) than the second one (static
frictional force) in order for $\sum F_x >0$ and thus it's
accelerating.

Again the forces are equal and opposite per Newton's third law. You need to apply Newton's second law individually to the car and the ground to determine the acceleration of each.
The static friction force is the only external force acting forward on the car. Then, per Newton's second law (neglecting air resistance and rolling resistance) the static friction force causes the car to accelerate forward with an acceleration of $a_{m}=F/m$ where $m$ is the mass of the car.
Per Newton's second law the force the wheel exerts backwards on the ground applying a torque on the Earth giving it an angular acceleration $\alpha$ backwards equal to $\alpha=F/Mr$ where $M$ is the mass of the Earth and $r$ is its radius. Since the product $Mr$ is so large, its angular acceleration is infinitesimal and therefore it appears stationary.
Hope this helps.
A: As the car accelerates the wheels rotate faster.  Friction opposes this increase in rotation where the tire contacts the road and pushes the tire hence the car forwards.  If there is no slip of the tire on the road, the frictional force is static.
A: It's true that static friction from the road is the force that accelerates (or helps decelerate) a car, but there is also “rolling friction” which results from the deformation of the tires (and sometimes the surface beneath them).  (Its like the car is going “uphill”.)  This must be overcome to accelerate, and with the engine “out of gear” this combines with friction from the air to slow the car.
