I feel like a lot of these questions are asking the same thing, so do tell me if I've misinterpreted any.
1. Is the magnetic dipole picture of assuming a circular loop carrying a current is a kind of "approximation"? Or in other words, is the field determined by assuming magnetic fields due to magnetic monopoles different from the original field due to circular loop?
Yes. For example, as you can clearly see in the figure, the two fields don't even point in the same direction in the interior.
In short, how accurate is the magnetic field determined by assuming a circular current carrying loop as a magnetic dipole?
It gets more accurate the further away you get, because both of them approach the ideal dipole field, $$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4 \pi} \left(\frac{3 \hat{\mathbf{r}} (\mathbf{m} \cdot \hat{\mathbf{r}}) - \mathbf{m}}{r^3} \right).$$ More precisely, at large distances, both of these fields differ from the ideal dipole field, in both magnitude and direction, by a fractional amount proportional to $\ell/r$, where $\ell$ is a characteristic length scale. For the loop picture, $\ell = \sqrt{A}$. For the pole picture, $\ell = d$.
As you go to smaller $r$, the divergences get a lot bigger, e.g. for $r \lesssim \ell$ the fields are completely different from the ideal dipole field, and from each other.
1. Does the pole picture causes variation only in the direction or also includes difference in the magnitude of the field?
Yes. 780
For example, as you can see from the figure, the magnitudes diverge at the poles and at the loop, and there's no corresponding divergence in the other picture.
1. Where does the pole picture give accurate result for both magnitude and direction of the magnetic field around a circular loop carrying a current?
At large $r$.
1. What are the specifications for the choice of $m$ and $d$? From $md=iA$, the product $md$ is a constant for a particular $i$ and $A$, however, we're free to choose $m$ and $d$ in such a way it satisfies the condition. So does a small value of $d$ (and large value of $m$) give more accurate results, or is it the other way round?
What is exactly true is that in the limit $d \to 0$ and $A \to 0$, the two fields approach each other, because they both become the same as the ideal dipole field.
When $A \neq 0$, it's not clear what value of $d$ gets a more "accurate" result, because it depends on how you define "accuracy". For example, if you wanted the field at $r = 0$ to be as accurate as possible, then you should send $d \to \infty$, because that gets zero field at $r = 0$. But then that would get the field at $r \approx \sqrt{A}$ totally wrong. If an IIT JEE problem asks you what the most "accurate" result is, this is a vague and undefined criterion and the correct answer is whatever random thing the test writer had in mind at the moment. 780