How does electricity pass through a surface (or more generally, conductive material with complex geometries)?
Part 1. 

The current will distribute through the disk proportionally to the resistance of each of the infinite paths it can take. Essentially the disk is a whole bunch of resistors in parallel. In order to work out the effect, you would have to sum the currents through each possible path by integrating current over the 3D structure.

In terms of whether anything changes when moving the connections, yes it will. If you consider both wires placed on the circle, all of the current paths have to travel at least the distance between those connections through the structure. If both are placed at the centre of the circle, the shortest current path will be only the thickness of the disk. Conversely if you place one at each end of a diameter line, all current paths will be through at least a diameter length of the material.

Clearly if you change the length of the current path through any material, the resistance of that path will change. The longer the path, the higher the resistance. If the total resistance in a circuit is increased without changing the supplied voltage, this will the lower the current.



Part 2.

If a person touches the disk and no closed loop is formed, then no, it will not change anything. However this is not physically possible.

In practice the finger will have some conductivity and be touching more than just an infinitely small point. As a result of the contact point being distributed across an area of the disk, it is equivalent to adding an additional resistive path into the circuit which will cause a small current change by reducing the resistance of the disk.

Additionally, if there is any AC component to the signal, the finger would act as an additional capacitance which will have an effect on the impedance of the disk.