Why does an object of constant velocity come to rest? As far as I understand, friction is supposed to be present only when there's a force pulling an object forwards on a plane.
So, for example, an object would be lying on a horizontal plane; a force is exerted on it and as such a force of friction is produced. The force is shortly after removed, and at that time it would've accumulated some velocity since it was being accelerated by an unbalanced force. What would happen to that velocity? Would the object come to rest?
Does this mean that the force of friction is still present after removing the pulling force?
And what is the value of that friction? (Is it $ \mu R$, or less?)
 A: I think you are confused about differences between the static friction and kinetic friction.
OBJECT IS AT REST
When an object is at rest, there is a static friction on the contact surface
$$F_{fs} = \mu_s n$$
where $n$ is magnitude of the normal force, and $\mu_s$ is coefficient of static friction. In order to move the object, the resultant force from other forces must overcome the static friction force. As long as the resultant force magnitude is less than static friction force, the object remains at rest! In order for the static friction force to satisfy the first Newton's law, it is actually defined via the $\min$ function, but the idea remains the same.
OBJECT IS MOVING
The moment object starts moving, i.e. as soon as velocity is not zero, the static friction disappears and the kinetic friction starts acting on the object, always in the opposite direction to the motion:
$$\vec{F}_{fk} = -\mu_k n \hat{v}$$
where $\mu_k$ is coefficient of kinetic friction ($\mu_k < \mu_s$), and $\hat{v}$ is the velocity (motion) unit vector. It takes greater external force to get the object moving than to keep it moving at the constant velocity.
FRICTION FROM THE WORK PERSPECTIVE
For a constant force $\vec{F}$ acting along some distance $\vec{x}$, the work is defined as a scalar product between the force and the distance:
$$W = \vec{F} \cdot \vec{x} = |\vec{F}| |\vec{x}| \cos\theta$$
where $|\vec{F}|$ is a magnitude (length) of a vector and is by definition always positive, and $\theta$ is the angle between the two vectors. Since the kinetic friction force acts always in the opposite direction to the motion, the angle in that case is $\theta = 180^\circ$ and the work done by the kinetic friction force is always negative! Also note that the static friction force does no work because the distance in that case is zero.

The force is shortly after removed, and at that time it would've accumulated some velocity since it was being accelerated by an unbalanced force. What would happen to that velocity? Would the object come to rest?

Yes, the object would come to rest eventually, which is evident from the work-energy theorem:
$$K_1 + W_\text{other} = K_2$$
where $W_\text{other}$ is the work done by external forces. Since the work by the kinetic friction force is always negative, if that is the only force acting on the object then the final kinetic energy $K_2$ will become zero after some distance $x$. At that moment the kinetic friction force stops acting on the object and there is again the static friction force which does no work.
A: The object will come to rest. Friction is "always" present. There is no frictional force if the object isn't currently moving, but when you exert a force or if the object is moving, there is a frictional force.
The value of that friction is $\mu R$ where $R$ is the reaction force.
A: Friction is not about other forces. It is not even just about speed. It is about sliding. Technically, that means that it is about relative speed. Solely and purely.
Friction comes into existence in order to stop sliding. And that is why a moving object that slides over the ground comes to rest. A moving object in space wouldn't come to rest as it wouldn't feel any friction because nothing is sliding over something else.
Note that there are basically two types of friction:

*

*Kinetic friction which appears during sliding and wants to stop it (when a surface moves over another)

*Static friction which appears when external forces try to initiate sliding and wants to prevent it (e.g. when pushing on a table without it moving)

You are in your question referring to kinetic friction because you mention motion (sliding, relative velocity). But in any case, the statement that Friction comes into existence in order to stop sliding is true for both types.
A: I have always, always hated how friction is explained at the school level.
The issue you are facing is that all these shenanigans about static friction, kinetic friction and even the misnomered monstrosity the rolling friction is are highly empirical. That is, given Newton's laws of motion we try to describe how objects around us behave, almost completely forgoing why they do it. And like many other forces we deal with in mechanics, we do not really concern ourselves with how these forces came to existence: instead, we proclaim: "observe, it quacks like a duck, it must be a duck: the friction force is linear, static friction is greater than kinetic friction etc.". While we are at it, the coefficient of friction can be greater than one, although some textbooks/teachers would claim otherwise: it is sometimes harder to push an object than to lift it.
This barely touches on the understanding of how these forces are produced, and I have found myself and many others utterly puzzled by these purely empirical explanations, especially if done without accompanying experiments.
Teachers would normally say "friction is always present" and when you raise an argument of the body at rest on a horizontal plane, they would say it is present but equal to 0. This is correct from a physics/math perspective (it is convenient to think of it as a lower limit), but a bit nonsensical from a layman perspective: when you have zero apples, you don't "have apples, just their quantity is zero".
To help understanding all that mess, I suggest considering underlying mechanisms for friction at a microscopic or, at least, mesoscopic levels. Static friction has a lot in common with the normal force and is also governed by elastic deformations. Consider breaking a biscuit in half: does it make sense to you that it pushes back on your finger exactly as strongly as you push down on it until it cracks? The same thing happens with friction on a microscopic level: all these small bumps push at each other until something breaks. In that sense yes, friction is always present - a solid body is always ready to push back at you, so regardless of the external force you have applied and removed to a microscopic body these small bumps on contacting surfaces still hit and deform each other and exert forces on each other - macroscopically, it manifests as deceleration corresponding to the kinetic friction force ($\mu\vec N$, where $\mu$ is found empirically: people might list dry friction $\mu$ and lubricated friction $\mu$ separately - think of it not as of fundamental principle in physics but as of engineering look-up table).
In other words, if you want to understand how it works, think of microscopic level. Friction in the mechanics course is instead driven by engineering problems like "how hard have we tug on that rope using a slope and a block to get the dang thing moving".
/rant
A: 
... friction is suppose[d] to be present only when there's a force pulling an object forwards ...

Incorrect. Friction always acts on objects to oppose relative motion between two surfaces, regardless of whether a force is or is not applied to one or both objects. If two touching objects are in motion relative to one another then kinetic friction will always act so as to reduce their relative speeds.
