firtree is correct - I will just try to flesh out his answer a bit.
(1) Your last question first - charge (or current) at a point is like mass at a point.
For finite masses, if you want to see how much is contained in an infinitely small volume (i.e., at a point), the answer is zero. So instead, people consider the mass density which can have non-zero values at a point. You probably understand the relationship between mass and (mass) density quite well.
Similarly for a finite current, the amount of current at a point (i.e., in an infinitely small volume) is zero. The current density is the limit of the amount of current in a small volume around a point as the volume goes to zero - just like mass density, but with current instead.
So, just as one speaks of mass density at a point and not mass at a point (for extended bodies), one speaks of charge density at a point and not charge at a point or current density at a point and not current at a point (we're ignoring point particles for now - they do fit into this formalism, but you need Dirac delta functions).
(2) Now, in analogy with mass flow, your picture of flow of charge is correct. Mass density times velocity gives a mass current density. $\vec{j}_{m}(t,\vec{x}):=\rho_{m} (t,\vec{x}) * \vec{v} (t,\vec{x})$. If you have a mass current density $\vec{j}_{m}$ and want to know the mass flow $\dot{m}$ through some area A, then you take
\begin{equation}
\dot{m} = \int \vec{j}_{m} \cdot d\vec{A}
\end{equation}
Similarly, charge density times velocity gives a charge current density. $\vec{j}_{q}(t,\vec{x}):=\rho_{q} (t,\vec{x}) * \vec{v} (t,\vec{x})$. If you have a charge current density $\vec{j}_{q}$ and you want to know the flow of charge $\dot{q}$ through some area A, then you take
\begin{equation}
\dot{q} = \int \vec{j}_{q} \cdot d\vec{A}
\end{equation}
So, the picture in your head is quite close - just picture charge or a fluid of charged particles flowing.
1) What is the 'mobile' surface charge density? Isn't the surface
current density itself the 'mobile' surface charge density?
Not all charges are mobile. For instance, protons and most electrons in a solid metal are not mobile. So, here, surface charge density could be interpreted as a surface density of free electrons.
2) Is the small current dI in the whole ribbon or only part of the
ribbon? I'm guessing its the whole ribbon of width dl but I want to
make sure.
Yes, dl is the width of the whole ribbon.
3) ...Is the current flowing only on the surface of the wire or is it
flowing naturally through the whole volume of the wire and I need to
only account for the current on the surface?
The current density is not always uniformly distributed through the whole volume of a conductor: most of a high frequency AC current, due to the skin effect, flows in a thin layer under the surface of a conductor. In such cases, it makes sense to talk about a surface current or a surface current density.
I'm able to derive K will be equal to I/2πa mathematically, but if the
current is flowing throughout the volume of the wire, how can I
visualize the unit length perpendicular to that flow?
As mentioned earlier, the current can flow mostly along the surface of a wire, in which case, the conventional current density, a current through a unit area of the wire's cross-section, could be replaced by the surface current density, a current through a unit length of the wire's circumference.
Best Answer
The current is due to moving charges (electrons).
Imagine 9 electrons are moving into the paper (or screen)
Diagram A has a low current density, diagram B has a higher current density.