Can a voltage regulator work below minimum Vin?
In general, below the minimum Vin, linear regulators present themselves simply as a saturated transistor. 

The output then simply follows Vin minus the saturation voltage of the pass transistor/darlington.



Of course there is some minimum voltage needed to turn on that transistor. 

Some devices also detect and shut down when the input is too low. 

However, the older 78xx series regulators however are pretty basic and will continue to pass Vin down to a diode drop or two above ground. Vin needs to be higher than Vbe of Q17 and Q16 + VSat of Q9, so about 1.8V or so. Below that the transistors will only be partially on.

Interestingly the current limiter Q15 should still work when it is in bypass mode.

Here is a quick simulation with an LT1084-5 LDO 5V regulator into a 1K load.



Note above about 1.9V the output follows Vin minus about 1V. Below 1.9 it's more of a resistor.

Here is the same device with a 0.1 Ohm load.



Note the current limit kicks in at 6A with Vin merely ~2.5V.