Intuition on Magnetic Dipoles and Magnetic Moment 1) Magnetic Moment in terms of magnetic poles:
According to Wikipedia, " Magnetic poles always come in pairs, their forces partially cancel each other because while one pole pulls, the other repels. This cancellation is greatest when the poles are close to each other i.e. when the bar magnet is short." 
My interpretation: Since 'the cancellation is greatest when the poles are close to each other', practically no force will be experienced when the the distance between the poles tend to zero. Therefore, no field lines will emerge, correct? (Though I know there is something terribly wrong with this)
2) Magnetic Dipole as limit of current carrying loop:
"A magnetic dipole is an idealized current loop - where area goes to zero and current to infinity so that their product is finite. The closer you approximate this situation, the more the field starts to look like a dipole field - analogous with the field created when you approach a positive and negative charge closer and closer together, with the product of charge and distance (the dipole moment) constant." 
(a) Why area zero and current infinity? What good is that?
(b) When positive and negative charges approach each other, the distance between them changes. So, how can the product of charge and distance be constant ?
(c) This image again contradicts my interpretation of Magnetic Dipole Moment.

 A: Let me quote the sentences from the beginning of the paragraph about the Magnetic pole representation (boldly highlighted by me):


The sources of magnetic moments in materials can be represented by poles in analogy to electrostatics. Consider a bar magnet which has magnetic poles of equal magnitude but opposite polarity. Each pole is the source of magnetic force which weakens with distance.


The last sentence is a simplification, and not very helpful one I think.
A permanent magnet has a magnetic dipole moment due to the permanent alignment of the magnetic dipole moments of the involved subatomic particles (mostly electrons). Now imagine that you have two bar magnets with the same number of aligned electrons, one of them thin and long and the other bolt and short. Observing their field lines one can see that from the bold magnet nearly all field lines emerge from the end surfaces. From a thin and long bar magnet a lot of field lines emerge from the cylindrical surface.
 On the sketches the first shows how field lines are going for a bold and short magnet, the second sketch shows that for a thin and long magnet.
This has to do with the magnetic saturation of the material and this saturation has to do with the availability of alignable subatomic particles.

The closer you approximate this situation, the more the field starts to look like a dipole field - analogous with the field created when you approach a positive and negative charge closer and closer together, with the product of charge and distance (the dipole moment) constant."
(a) Why area zero and current infinity? What good is that?

Such an approximation is not permissible. In a zero volume there aren’t any electrons with their magnetic dipole moments and hence there isn’t any magnetic field.
Edit after a question in the comments

Does this mean that the dipole moment is large if the surface area of the poles is larger? (Since larger surface means more aligned subatomic particles)

The strength of a permanent magnet depends from the amount of aligned subatomic particles and the field lines which represent the magnetic field at any point inside and outside the magnet are closed loops. Imagine that only a certain number of lines (as a model of the field) are able to be in some cross section of the bar magnet. This saturation limit depends from the available subatomic dipole and their mobility to be fully aligned. Once the saturation is reached, the field lines are going out sideways through the cylindrical surface of the magnet.
So to get a stronger magnet one can use a larger volume and one can use rare earth magnets with their large magnetic dipole moments in each atom.
A: The answer of both questions lies on the idea and concept of multipole expansion. The idea is basically the same as simple common Taylor series expansion, applied on the size $d$ of the source distribution.
First take electric multipoles as an example. You have the total field being the sum of zeroth order terms, first order terms, second order terms ... in $d$, which are, respectively, called the monopole, dipole, quadrupole,... and so on.
The classical model of a dipole consists of two charges $q$ and $-q$ separated by a distance $d$. Why it is $q$ and $-q$? The reason is that the monopole moment is just the total charge $Q$, and if non-vanishing, will be the dominant term. To kill this term, you must have a system with total charge being zero.
So now the dominant term is the dipole term, which goes like $qd$. But there are also higher order terms. For example, the quadrupole term (which happen to be zero in this case) which goes like $qd^2$, the octupole term, which goes like $qd^3$ and so on.
So how to make a pure dipole? You can let $d\to 0$ and $q\to \infty$ so that $qd$ remains finite. In such a way, the higher order terms $qd^2$, $qd^3$, ..., all go to zero.
The magnetic case is similar to the above.
So to answer your questions:

My interpretation: Since 'the cancellation is greatest when the poles are close to each other', practically no force will be experienced when the the distance between the poles tend to zero. Therefore, no field lines will emerge, correct? (Though I know there is something terribly wrong with this)

The size of the dipole goes to zero but the strength of the source goes to infinity. The dipole field goes with the product of source strength and size, and hence tends to a finite non-zero value.

(a) Why area zero and current infinity? What good is that?
(b) When positive and negative charges approach each other, the distance between them changes. So, how can the product of charge and distance be constant ?

(a) Area going to zero kills the higher order terms.
(b) You do not decrease the separation $d$ between fixed charges $q$ and $-q$. Instead think of it in this way, if you have a system with charge $q$ and separation $d$, then the error terms goes like $qd^2$, $qd^3$ ... and so on. Now construct another dipole with charge $q/10$, separation $10d$, which maintains the same dipole term. But then errors becomes $dq^2/10$, $qd^3/100$, and so on. Now construct yet another dipole with charge $100q$ and separation $d/100$, then the dipole term remains the same but the higher order terms become $qd^2/100$, $qd^3/10000$, and so on. Under such a limit, you will have a pure dipole.
