Measuring voltage across short circuit
Assume a simple short circuit with the probes on a multimeter at any point along the circuit:


OK, but the 5 V supply will have to go into current limit. You can't have 5 V and a short circuit.


  Can we assume that the voltage read will be 0 V, as all points along the wire have the same voltage?


Yes.


  If we then add a resistor and measure the voltage across the resistor, at which resistance would you register 5 V?


It depends on the current capability of the power supply. For example, if it can supply 2 A at 5 V then ...


  How does Ohm's law verify this?


... \$ R = \frac {V}{I}  = \frac {5}{2} = 2.5 \ \Omega \$ and that's the minimum circuit resistance that the PSU can drive while maintaining 5 V.

It always works.



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  What's confusing me is how the pin at GND suddenly reads 0, even as the resistance > 2.5 increases? Why wouldn't it read it with a lower resistance, even if the circuit was "demanding" more current than the PSU could handle?


GND in a circuit is just a reference point. You can think of it as where we would normally connect the black lead of the multimeter which is switched to V DC. Since the wire in question is thick and made of a good conductor then all points along the wire will be at ground potential when measured with the red probe of the meter. In a real circuit with current you would get an increasing millivolt reading the further you move the red from the black lead.



Figure 1. A rheostat. Note the wound resistance wire and the exposed strip with the movable contact. Source: Resistor Guide.

Now if we add in the resistor the situation changes. The current is reduced and if we were able to touch the red probe at various points along the resistor we would see the voltage rise to 1 V at 20% up from the bottom, 2 V at 40%, etc. until we reached the top and we would measure 5 V. We could use the rheostat in the photo by connecting one end to +5 V and the other to GND. Then connect the red probe to the wiper and move it along.


  Really, what I'm asking is, if current is the same throughout the circuit, how does one pin know it is upwards of the resistor, and therefore read 5 V, and the other downwards, and read 0 V. I ask this in theory and practise.


The pin doesn't know but the voltmeter can measure it. The potential difference with GND is decreasing as we move down along the resistor.