The current direction along the coil depends on the sense of the winding (or thread), so it can be either way. (I can't really make sense of the perspective in your drawings.)
The right-hand rule for coils and their magnetic field implicitly assumes that the current flows on circles around the solenoid axis. In reality, these are of course very slightly bent and form a helix, and the helix can be left-threaded or right-threaded. This does not apreciably change the way the current flows in the angular direction (tangential to the solenoid), and thus the magnetic field. However, the net current direction is reversed.
So if the solenoid is, say, vertical, and magnetic north is up, then the current (direction chosen as for the right-hand rule) is flowing up for a right-winding coil, and down for a left-winding one (if I didn't confuse myself here).
For the physical interpretation of the limit: imagine you had a current loop whose size you could decrease easily, like pulling on a drawstring. If you just make it smaller, you decrease the magnetic moment $\mu = IA$; to keep the field the same, you'd have to increase the current. If you cut the area $A$ of your loop in half, but doubled the current $I$, you'd have the same magnetic field far from the loop.
The ideal dipole source has zero size and infinite current, and the ideal dipole field is therefore infinitely strong at the origin. That's annoying. But we have the same problem with the monopole field, like the electric field from a point charge, which is proportional to $1/r^2$ and is infinite at the origin. For the electric field we get around this by inventing quantum mechanics and discovering that a "point charge" is not actually a thing that exists. The proton has a finite size; while the electron is a structureless "point particle," for computing its electric field you actually care about the finite charge density described by its wavefunction.
I might rephrase your second question as "what do we mean by 'large distance'?" Suppose you have a real cylindrical solenoid, made out of wires, with length $L$ and radius $R$. The dipole approximation is only good if your distance $r$ from the solenoid is much larger than $L$ or $R$. If you're inside the core of the solenoid you see a uniform field; if you are a tiny gnat tunneling through the wall you might prefer to treat the local field as due to a locally-flat sheet of current. The local field is complicated.
For a complicated field, it's helpful to describe it using a multipole expansion. I've already hinted at this by reminding you about the monopole field, which is produced by a point charge, and gets weaker like $1/r^2$ as you move away. If two opposite-sign charges are near each other, their monopole fields approximately cancel, and most of what's left over is described by the dipole field. The dipole field gets weaker with distance like $1/r^3$ — that's sort of what we mean when we say that the monopole fields approximately cancel out. The dipole field also has the more complicated shape, which leaks information about the orientation of the charges at the source.
Two back-to-back dipoles also approximately cancel out. What's left there is called a "quadrupole field," which gets weaker like $1/r^4$ and has an even more complicated shape than the dipole field. There's an infinite series of these higher-order corrections, which get weaker more rapidly as you move away.
A current loop with radius $R$ produces a magnetic field with nonzero dipole moment, but also nonzero quadrupole, octupole, hexadecapole, etc. moments, all of which are parameterized by $R$. If you move from $r$ to $2r$, the dipole field gets weaker by a factor of $2^3=8$, but the quadrupole field gets weaker by $2^4=16$. If you move many $R$ away (or equivalently, rebuild your current loop so that $R$ is very small), eventually only the dipole shape of the field will be measurable.
Best Answer
If possible do as @AccidentalFourierTransform explained in a comment, namely:
Be aware, that some compasses are embedded in an oil capsule, so disassembling will destroy them mostly.
Also remember that the needle uses always both poles, as the magnetic field influences the structure of the needle as a whole. Otherwise it wouldnt't work, so the easiest is to use your imagination.
Perhaps you meant the declination which must be adjusted on the compass, which can influence the way you've got to take, when hiking close to the north or south? Navigation 101