Actually, all of the three questions are directly related.
As you noted, the Earth's magnetic field is generated by flow of molten conductive materials (probably mostly iron) in the outer core. (Contrary to popular belief, only the outer core and the oceans are liquid. The crust, mantle, and inner core are all solid. Solids can flow, but they have a shear strength. Liquids have no shear strength.)
However, this flow is turbulent.
Therefore, Earth's magnetic field can be described by having two poles, but in reality, it's much more complex. Part of this is due to crustal structure (i.e. the magnetic anomalies we typically work with in geophysics to, say, locate an igneous intrusion or understand the evolution of the oceanic crust). However, the part that's due to crustal structure doesn't change rapidly and only influences a very small region. The "longer wavelength" (i.e. broader) differences are due to variations in the flow of the outer core.
Every "eddy" (i.e. any vorticity) in the flow produces its own magnetic field. The sum of these is what produces the global magnetic field. At the surface, we're a significant distance from the flow that produces the magnetic field. Therefore, it gets "smoothed out" and these small perturbations are less apparent. It looks fairly close to a single magnetic dipole because the bulk of the flow is approximately around Earth's rotational axes.
This is the answer to your first and second questions.
The magnetic field is not the result of a single "current" of laminar flow, but rather the sum of much more complex turbulent flow. Therefore, it doesn't exactly match the rotation axes, and the position of the magnetic poles changes over human timescales.
I'll hold off on answer the third question, as you split it into a separate question, but basically, the flow tends to self-organize, producing a strong magnetic field that's close to a single dipole. Then, for reasons that aren't entirely clear, it episodically de-stabilizes (basically, you'd have multiple equally strong poles that somewhat cancel each other out), and then re-stabilizes (sometimes in the opposite direction).
If you're interested in a "pop-sci" style writeup, there was a very nice Scientific American piece on this awhile back written by Glatzmaier (a very big name in the field). Unfortunately, the web version is missing the figures that were in the print version, but they're mostly from Glatzmaier & Roberts, 1995 or one of the follow-up papers by the same authors.