Why do we feel a force in circular motion? I'm wondering why we only feel the centrifugal force in a circular motion. 
I did look at Why do we only feel the centrifugal force?  question which is exactly that, but I'm not really satisfied with the answer. When you are rotating in a non-inertial reference frame the centripetal force pulls you towards the center, but this acceleration is balanced by the centrifugal force. But if that's the case, why do we feel a centrifugal force? 
The other question says that it's a reaction force, but since the acceleration is balanced, why do we still feel this force. And in a related example, if a ball starts to spin in a circle, why doesn't it keep the same radius, but instead is pushed out because of centrifugal force? Aren't the centrifugal force and centripetal force balanced?
 A: When you're driving a car (United States car) and make a fast right turn, you indeed feel both a centrifugal and a centripetal force in your noninertial reference frame. 
The centripetal forces are the friction underneath you with the seat as well as the normal force from the door on your left both pushing you to the right (toward the center of the circular motion). The friction with the seat, since it doesn't act on your center of gravity, actually applies a torque to you that causes you to rotate counterclockwise. 
Meanwhile, the centrifugal force balances the centripetal force and acts on your center of gravity, causing you to stay in your seat and not be thrown to the far right passenger side of your car by the centripetal forces (assuming you're not wearing a seat belt of course!).
Why do you still feel the force even though it's balanced? You feel two things. One is the unbalanced torque on your body that causes you to lean to your left. The other is the centrifugal force and the normal force of the door compressing your body. Balanced forces acting on “different sides” of your body can still be felt as compressing your body. These are the things you feel when in circular motion in a car.
If a ball is rolling around in a cone, if it goes faster (perhaps driven by a propulsion fan of some sort) it will move outward and upward. This is precisely because the centrifugal force is not balanced by the centripetal force!
A: Centrifugal force is a fictitious force or hypothetical one or not applicable physically, then how it can balance any force.
For circular motion, haven't you heard about ' inertia of direction' ?
A: The thing is that net force of $0$ does not mean "no forces act on you". Indeed, you feel pushed into your chair as you sit on it, and if you were sitting in your chair on an upward accelerating elevator you would feel pushed into your chair even more. In either case the net force acting on you in your non-inertial frame is $0$ (once you take pseudo-forces into account), but you would "feel" more in the case of the upward accelerating elevator compared to just sitting in a chair on the ground.
This shows us the key to your question. We "feel in the opposite direction of acceleration (as observed from an inertial frame)." In the case of the elevator, we more we accelerate upwards, the more we feel pushed downwards. In circular motion, the acceleration is inwards, so we feel "pulled" outwards. 
You can essentially link this to Einstein's equivalence principle: experiencing an acceleration is just like being in a gravitational field pointing in the opposite direction as the acceleration. Indeed, this is even why you feel pulled to the ground right now, you could argue that you are actually being accelerated upwards (approximately).

Another way to think about it. In non-inertial frames Newton's laws no longer hold. But which law/laws does/do not hold? Well, technically speaking Newton's second law no longer holds. In other words, we can feel forces without experiencing accelerations in our non-inertial frame. So it should be no surprise that we still feel an outwards force when we don't observe acceleration in our frame, if we felt nothing this law would hold!
However, this idea doesn't sit well with us. What if we still want Newton's second law to hold? Well then we introduce the idea of fictitious forces. However, by doing this we have now thrown Newton's third law out the window. Forces now longer have to have an equal and opposite partner. Forces do not have to arise from interactions anymore. So your claim that the centrifugal force is a reaction to the centripetal force is actually false! Remember, action-reaction pairs do not act on the same object. The centrifugal force arises from the non-inertial reference frame.
So I would argue that while we have introduced fictitious forces to help us keep N2L valid, I think it is better to say that we are "feeling" the effects of being in a non-inertial reference frame. 
