Does an action-reaction pair always contain the same kind of force? According to Newton's third law, the forces in an action-reaction pair must have the same magnitude and opposite directions. But do they have to be the same kind of force (gravitational, electromagnetic, strong, weak)?
 A: There is no such thing as "the force of A on B." Force is a thing that only exists between objects. That's all there is to it.
An important note is that force is not what physicists consider an observable. That is, you cannot directly measure a force. In classical physics, the only observables are distance, mass,  and time. In this context, we infer the force on an object by measuring its mass (observable) and acceleration (length and time observables), and then applying Newton's 2nd law.
Danger: Extremely esoteric stuff below...
An interesting philosophical detail here is that we then need to determine mass without using a force (such as gravity, spring, etc.). If we accept the equivalence of gravitational and inertial mass, then we can measure the mass of an object through a collision with another object of known mass.
Addressing comment below
The justification is the following: From a practical point of view it makes total sense to say "the force of A on B." For instance, an engineer might want to know the force a bushing puts on an axle. Or you might want to calculate the torque on a compass needle from an external magnetic field. It would be dumb to ask an engineer to calculate the mutual action between an axle and its bushing. However, the OP asked a deeper, more philosophical question that requires a subtler and more nuanced answer. This requires pulling away the shroud and pointing out that an N3 force pair does not actually consist of two forces. There is ONLY a mutual force. Thus the concept of the equal and opposite forces being of a different nature is nonexistent.
A: Yes, and it is one of the “tests” that one can use to check if two forces are a Newton third law pair.  
The pairing always exists.  
A body on a table has two forces on it.
The contact force on the body due to the table and the gravitational force on the body due to the Earth.  
Although these two forces have the same magnitude and are opposite in direction they are not a Newton third law pair for the following reasons.
They act on the same body, Newton third law pairs must act on different bodies.
One is a contact force and the other is a gravitational force.
Take the table away the contact force disappears but the gravitational force acts on the body.  
The Newton third law pair to the gravitational force on the body due to the Earth is the gravitational attraction on the Earth due to the body.  
The idea that the Newton third law pairing always exists cannot be satisfied if the forces are of different types.
A: There is no such thing as "kind of force" in pure Newtonian physic. A force is defined only by its intensity and its direction.
A: We're talking about the "two sides" of the SAME force, so it must be yes, they are of the same type.
For example, if I attract you gravitationally, then you are also attracting me gravitationally.  There is only one physical source of the force (in this case, gravity) and it is pulling us both equally and oppositely.  Since there's only one thing, it has, and can only have, one type.
Or, to put it another way, according to Newton's third law: When I push against you, you are also pushing me back.  It is the same physical thing doing both pushes, in this case electrical repulsion of our atoms.  
Or like this: Imaging I support a book of 1N weight in my hand (against earth gravity).  I can't support that book without supplying 1N of force to stop it falling.  And that force will be provided by the electrical repulsion of the atoms in me vs the atoms in the weight.  The book feels that 1N force upwards, and I feel it downwards on my hand.  There's only one force doing that.  There's ALSO a gravitational attraction pulling the weight towards the earth, balanced exactly by the gravitational attraction pulling the earth towards the weight.  So there are (at least) two instances of the third law in play.
A: If you get down to the fundamental layer, what we perceive as Force on a macroscopic scale is really just exchange of field bosons. So when two particles interact via a boson exchange, it is the same force because it is mediated by the same boson.
The only way you could see asymmetric forces would be if the boson could oscillate into another form. For example, a proton could emit a photon (boson of electromagnetic force), intending to repel another proton electrically, but then the photon could oscillate into a graviton (gravitational force boson) and thus transmit a gravitational force to the second proton.
I've never heard of this happening...
A: For conservative system you get the force from this equation 
$$F_i=-\frac{\partial U} {\partial q_i}$$
$$F_j=-\frac{\partial U} {\partial q_j}$$
where U is the potential energy and q is the generalized coordinate, thus action force is equal reaction force. Between two parts you have the same potential energy like spring gravitation ...
Example:
$$U=\frac{k}{2}\left(x_1-x_2\right)^2\implies\\[2em]$$
$$F_1=-\frac{\partial U} {\partial x_1}=-k\,(x_1-x_2)$$
$$F_2=-\frac{\partial U} {\partial x_2}=+k\,(x_1-x_2)$$
thus:
$$F_2=-F_1\quad\surd$$
Edit:
additional remarks:
every cut force (for example constraint or friction force or whats so ever)  ,must  obey 
Newton's third law ,if not your equation doesn't described correct the physical situation.
Let say you have two body ,  mass equal one , connected by a rigid  rod  ,  you get
the constraint forces $F_1$ and $F_2$  by "cutting" the rod. the EOM's are:
$$\ddot{x}_1=+F_1$$
$$\ddot{x}_2=-F_2$$
the physic "tell" us that if the rod is rigid, $x_1-x_2$ must be zero, thus  $F_2=-F_1$ this is Newton's third law action equal reaction
A: Whether or not the forces must have the same flavour is a choice of definition.
Two particles (A and B) exert a force on one another Newton tells us that the magnitudes of those two forces will be equal but their directions will be opposite. The question is, are the two forces of the same "flavour" (Electromagnetic, gravitational).
Suppose that their WERE two particles that did infact have an interaction where the a force on one from some source (say Electric) was matched by an equal and opposite force on the other from a different flavour (gravity). If this were ever experimentally observed one could draw one of two conclusions:


*

*The reaction force has a different flavour to the action force in this specific case

*The two forces we thought were fundamentally different actually have the same flavour.


Which one you pick is just words and definitions, their is no physics different between them. I think people would more likely take the second definition.
The same logic can be turned on its head. Imagine that someone who learned physics in a weird way tells you about two different flavours of force they learned about: one is "The negative E force" that pulls negative charges towards positive ones, and the other is a completely different force of a different "flavour" called "the Positive E force" that pulls positive charges towards negative ones. 
This person agrees with you about what happens when a positively charged particle interacts with a negatively charged one (they attract each other). The only difference is the language they use (how they cut up the "flavours").
A: Newton's third law says nothing about the type of forces involved, so in itself it does not require the forces to be of the same type; however, they must be because the four fundamental forces act independently of each other. For example, two masses will attract each other gravitationally, each having an equal but opposite impact upon the other. If they are both positively charged they will also repel each other electrically, and the repulsion will affect each of them equally.
