What is the exact role of friction in rolling without slipping? When an object rolls along a plane, we say that for the object to roll without slipping, the velocity of the center of mass must be equal to angular velocity times radius, so that at the point of contact with the ground the two velocities cancel each other, and the point is instantaneously at rest. 
I understand this. 
But then we say that if the ground is rough, there exists a friction on the bottom of the wheel. Which gives it torque. And this friction is responsible for the rotational motion of the wheel. Now if there is a external force acting on the center of mass of the wheel, then the only way the wheel will roll without slipping is if the tangential acceleration is greater than or equal to the acceleration caused by the external force at the center of mass. I don't understand why this is. 
First of all, if the point is momentarily at rest, how does friction know which direction to apply the force in? Secondly, if friction applies a torque, couldn't it be the case that the linear acceleration caused by the same is greater than the acceleration due to the external force? In that case also, shouldn't the wheel slip?  
 A: The part you understand is correct. Let's get to the rest of your question.

But then we say that if the ground is rough, there exists a friction on the bottom of the wheel.

Not ideally, no. Just like how there is no static friction acting on a book resting on a table with no other horizontal forces acting on it, there is no static friction force acting on the wheel if there are no other forces/torques trying to change the wheel's velocity.

Which gives it torque. And this friction is responsible for the rotational motion of the wheel.

This is invalidated by the above discussion. If the wheel is rolling then it will keep on rolling. Friction is not responsible for this, just like how it is not responsible for keeping the book at rest on the table.

Now if there is a external force acting on the center of mass of the wheel, then the only way the wheel will roll without slipping is if the tangential acceleration is greater than or equal to the acceleration caused by the external force at the center of mass.

Here is where friction now comes into play, just like how static friction would be required to keep our book at rest of we started to push on it. Although I must say your terminology confuses me here. The simplest way to think about it is just that the static friction has some maximum value. If friction needs to be larger than this maximum value to prevent slipping, then slipping will occur. Once again, just think about the book.

First of all, if the point is momentarily at rest, how does friction know which direction to apply the force in?

Friction opposes relative motion. It "knows" which direction to act because that is the direction that opposes slipping. Once again, think about the book.

Secondly, if friction applies a torque, couldn't it be the case that the linear acceleration caused by the same is greater than the acceleration due to the external force? In that case also, shouldn't the wheel slip?

This is confusing again. Just talk about forces. I see many people, including you now, getting things mixed up with comparing "accelerations due to forces and torques". 

You can work out the problem in more detail using Newton's second law. What is interesting is that depending on where you apply your force and the type of wheel you have the friction force could act either in the direction of the applied force or opposite the direction of the applied force to prevent slipping. I discuss this here and here
A: When a wheel is rolling without slipping at constant speed, friction has no role: The rotational and translational speeds are perfectly matched, and neither force nor torque from friction is needed nor available. 
To put it another way, friction is about force and torque. Those in turn are related to angular and linear acceleration, not velocity. No acceleration, no force and/or torque, no friction. 
Now, what happens if some (external) force accelerates the axle?
In the absence of friction, the forward linear motion of the wheel increases due to the force on the axle, but there's nothing that increases the angular speed of the wheel:  With that constant angular speed, the touch point will be moving forwards.
In the frictionless case, the wheel will start sliding forward on the surface. But that sliding is what friction at that surface can oppose with a fore that generates a torque to speed the wheel up.
Imagine a thrown bowling ball. At first it's sliding, and the friction acts to make it spin until it's rolling.
With a small acceleration, the wheel might not reach the point of sliding and kinetic friction. For small accelerations and large friction, it'll stay in the static friction regime. But that static friction will be generating a force in that same direction, causing the wheel to speed up it's rotation until it's rolling at the right speed to be rolling without friction.
