Which force makes the contact point of an object in rolling motion rise? When an object is rolling without slipping, the point at the bottom has no tangential velocity. When the point at the bottom rotates and rises, what force lets the point rise with tangential velocity from having no tangential velocity from before? I read that static friction is zero for rolling objects on a flat surface, so which other force could do this?
 A: Short answer: 
(Internal) centripetal force acts on all points because the body is rotating. This is what lifts the contact point.
Long answer:
Let us observe the bottom point of an extended circular body in pure rolling. 

Pure rolling is a type of combined translational and rotational motion. This means that all the particles of the body are both moving linearly and in a circle. Hence, every point on this body has two velocities, one equal to the velocity of the center of mass ($v_m$), and the other angular ($\omega$).
Let's now observe the bottom most point. We say it is 'at rest' because the vector sum of velocities is zero. $$v_m+(-\omega R)=v_m-v_m=0$$ since $v_m=\omega R$.
However, we know that all points of a body in rotational motion have an angular velocity. When an object is rotating, all its particles will feel an internal centripetal acceleration towards the axis of rotation. 
This must be true for the contact point as well. It has, at all times, an internal centripetal acceleration equal to $\omega^2 R$. This acceleration acts on this point at all times, whether it is at the top, side, or bottom. 
It is this acceleration that internally 'pulls' the bottom most point back upwards and allows the body to remain in rotational motion. 
Note: It's important to note that the reason the body doesn't 'break' is because all these internal forces cancel each other out. The way to not run into the confusion you're having is to think of centripetal force as a prerequisite to circular motion rather than an effect. If at any point the value of $\frac {v^2}R$ is less than the minimum required centripetal acceleration, the body will leave circular motion. In rolling, each particle always satisfies circular motion. So, all points in the body will have centripetal acceleration.
A: You have to be careful about two different ways of thinking about the object. 
In problems like this, the object is treated as a single rigid thing. You ask what are the force on the object to learn how it as a whole accelerates. The forces you consider are typically gravity, the force of the floor or inclined plane holding the object up, friction, and ropes attached to the object. 
But the object is also a collection of atoms. You can ask about the forces on each atom. The total force on the object is the total force on all of its atoms. 
It is important that the object is rigid. This means that neighboring atoms exert forces on each other that are just strong enough to keep in their places. These are called internal forces. Forces always come in equal and opposite pairs. If atom 1 pulls on atom 2 with force F, then atom 2 pulls on atom 1 with force -F. When you add up all the internal forces, you find that all the pairs cancel, and the total is 0. This means that the internal forces do not accelerate the object as a whole.
But the internal forces can accelerate one part of the object in one way, and another part in the opposite way. And this is what happens when an object rotates. Each atom travels in a circle. 
If the object was made of powder, each atom would fly off in a straight line. But the object is rigid. Atoms hold on to each other. Each atom has an inward, centripetal force. 
And this is the force that lifts the bottom atom. It is balanced by a similar force that pulls the top atoms downward. 
This post has more about internal forces. 
A: In rolling motion it is not friction that is zero. Work done by friction is zero because all the points are just momentarily in contact with the rough surface.
Technically, as you said, the points are lifted up the next moment they come in contact with the floor. So, no displacement occurs in the direction of frictional force.
I think your confusion is about the tangential acceleration of the rolling body. The frictional force acting along the tangent at the point of contact increases the tangential velocity when it leaves the surface 
Frictional force acts, and is necessary for pure rolling. If you try to let a ball roll in a smooth surface, it will slide rather than roll. A certain amount of frictional force is a prerequisite for rolling motion.
