When can free-floating electrical contacts become dangerous (e.g., start a fire)?
The answer is going to depend a lot on what you mean by dangerous.  You have to pick a pretty clear definition to measure against before you can answer this question.  So let's use EN61010 Safety requirements for electrical equipment for measurement, control, and laboratory use. General requirements. Because I have that to hand. Safety standards for other electronic equipment have similar requirements.

Shock risk

EN61010 defines something as hazardous live if the voltage is over 33Vrms (or 70Vdc) and either the current is over 500uArms (2mA DC) or there is more than 45uC stored in a capacitor.

Fire risk

The standard considers circuits of less than 30Vrms (or 60Vdc) and limited to low currents such that power is less than 150W, and well separated from other higher energy circuits to be a low fire risk.  Circuits which don't meet this level must use extra insulation against shorts, or a fireproof container etc.  This level is based on there not being anything particularly flammable inside the device where the potential source of ignition is, the levels would be lower around a pool of petrol or a load of dry paper.

Single Fault

This is a more complicated concept.  No one failure in the device should allow something dangerous to happen.  For example, if there is some mains wiring, and that is separated from the case only by an air gap, that is not OK. The wire could break loose, touch the case, and give the user a shock. You would have to add some extra insulation, or attach the case to a low-impedance safety ground. The point is that a 5V/1A logic supply is not automatically considered safe if it is derived from mains with a transformer and rectifier. To be considered safe, the transformer must meet certain standards so a failure could not result in the low-voltage side getting mains voltage.