Electrical current characteristics through conductor
... does the current saturate the entirety of the conductor's cross section equally?


In general, yes, but for AC funny things happen in large conductors and at high frequencies.


  Would it make a difference whether it was AC or DC?


Yes. See below.

AC

Straight from the mouth of Wikipedia:


  Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor, and decreases with greater depths in the conductor. The electric current flows mainly at the "skin" of the conductor, between the outer surface and a level called the skin depth. The skin effect causes the effective resistance of the conductor to increase at higher frequencies where the skin depth is smaller, thus reducing the effective cross-section of the conductor. The skin effect is due to opposing eddy currents induced by the changing magnetic field resulting from the alternating current. At 60 Hz in copper, the skin depth is about 8.5 mm. At high frequencies the skin depth becomes much smaller. Increased AC resistance due to the skin effect can be mitigated by using specially woven litz wire. Because the interior of a large conductor carries so little of the current, tubular conductors such as pipe can be used to save weight and cost.


DC

The DC situation is much simpler but can be affected by geometry. For instance, take a large flat and wide L-shaped conductor - i.e., cut from a sheet. The current will take the path of least resistance around the corner much like the racing line on a racetrack corner. You could expect the inside of this corner to run hotter than the rest of the conductor.



Speed of electron vs the speed of electricity

You haven't asked but ...

Have a look at the Wikipedia article on drift velocity. It shows the calculations for a current I = 3 amperes, and a wire of 1 mm diameter and comes up with the result that in this wire the electrons are flowing at the rate of −0.0000028 m/s. At this rate it would take 99 hours for an electron to travel 1 m along that wire! If we double the current the time taken is halved.

As the electrons move through the resistive material they interact with the atoms in that material and that's what limits the current.

With reference to Wikipedia's Speed of electricity: 


  The speed at which energy or signals travel down a cable is actually the speed of the electromagnetic wave, not the movement of electrons. Electromagnetic wave propagation is fast and depends on the dielectric constant of the material. In a vacuum the wave travels at the speed of light and almost that fast in air.


The speed of the electric current will be somewhere between 50% and 99% of c, the speed of light (\$3 \cdot 10^8~m/s\$). See Wikipedia's Velocity factor for more information.