question

In the manual, it states not to connect inverters to the load output. Inverters present a complex reactive load to an upstream switch. The upstream switch in this case is the MOSFET in the load output.

The load output does have over current protection which will helps to make the device short circuit proof. Bear in mind a short circuit is an over current event, not an over Voltage event and there is no mention of the load output handling over Voltage events.
A reactive load could produce a very high Voltage when switch off. This is known as flyback Voltage and can be hundreds or even thousands of Volts for a very short period of time, but long enough to destroy semiconductors in the blink of an eye. It is common for inverters to exhibit this and that is why the advice is given in the manual to not connect inverters to the load output.

I don't think the device has been specifically programmed to make up dangerous fault values, it's more to do with the device being damaged and it no longer has the ability to read the current anymore.
Where accuracy isn't critical, it is common practice to use the resistance of the MOSFET channel when it is on as a shunt to measure the current going through it. This is known as the RDSon of a MOSFET and is fairly constant in a working MOSFET and so the Voltage drop over the channel can be used to measure current with a fairly reasonable accuracy. In a melted MOSFET the channel resistance is likely to be extremely close to zero Ohms and so current measurements would also evaluate to zero Amps.
The high Voltage may have also continued is path of destruction further on.

So the answers are:
1) The load output does handle short circuit or over current events, but not over Voltage (flyback) events. In your case I suspect that the unit faithfully responded to an over current event caused by the inrush current to the inverter and switched the load output off which in turn produced a very high flyback Voltage from the inverter and destroyed the MOSFET which drives the load output. The failure mode of the MOSFET was most likely a melted silicon die. All three pins are now melted together and are all short circuited to one another. Drain and Source, the pins over which the current is measured are now reading as close to zero Ohms as practically measurable.

2) Given the likely failure mode of the MOSFET in its short circuit condition a shunt of what is now close to zero Ohms is not going to have a very high Voltage drove over it and so the microcontroller would only measure what is there. If it doesn't measure a Voltage then it would presume there is not current flowing as expected. It does not know that the shunt is melted.
So the firmware is working as expected, it has responded to the over current event and turned the MOSFET off, it shows that it has done so. It can't measure any current because the shunt has gone short circuit and so it shows zero Amps.
Now, even if the MOSFET has blown open circuit, no current flowing through it still, there is still no shunt to measure over. In this scenario the microcontroller would see full scale Voltage for the shunt measurement. This is also an expected scenario when the load output is switched off because the MOSFET channel resistance would be very high. It would need to be programmed to ignore this very high reading as unrealistic, especially when it has instructed the MOSFET to be off.
So if the microcontroller reads full scale at its ADC input and the load output has been driven off, then it would make sense that it reads zero Amps.
It's very unlikely that the microcontroller has an input to detect that the MOSFET has been destroyed.

Basically, it you use the device otherwise as instructed and it ends up getting damaged, then it would be expected that it gives erroneous readings and acts strangely if at all.