4- 20 mA Transmitter Signal Greater Than 20 mA

A 4-20mA output from a transmitter will always be in this range unless either the loop resistance it is connected into is too high to sustain the top end, or there is a fault of some kind in the transmitter.

The reason for a "live zero" at 4mA is simply to enable fault detection, the most common of which would be a transmission loop going open-circuit for whatever reason. However, an inadvertent short circuit in the loop part-way between transmitter and receiver will have the same effect in the receiver as any other reading of less than 4mA.

<...-3000 C (too cold) or +3000 C (too hot)...>

Such indications are clearly nonsense, as

1. the Celsius scale does not go below -273.15, and
2. most common materials are either molten, vapour or the products of combustion at the upper figure.

Although humans know this from everyday experience and from training, the equipment can detect and take action very quickly if nonsensical readings are received, provided it has been correctly programmed by those humans.

<...what will be the expected output or reading at the controller? ...>

The receiving equipment will be programmed for a defined input scale to match the range of the sending scale on the transmitter. It is commonplace these days to program the analog inputs of the receiving equipment to make it display an out-of-range fault alarm of some sort when the current received is either less than 4mA or greater than 20mA by a little amount, and to program further to invalidate the received signal within the receiving equipment so as to mitigate the effects of this loop fault. Examples:

• Say the range of the 4-20mA were 0-100 units. An input of 22mA could give "100 and over-range fault" in the receiving equipment, rendering the 100 invalid, and an input of 3mA could give "0 and under-range fault" rendering the 0 invalid using this set-up.
• An invalid signal can also be programmed to freeze the last valid reading as the display or the control measurement for other functions within the receiving equipment, determination of which depends entirely upon the application.
• Further, the inputs of several sensors can be compared in "two-out-of-three" or even "three-out-of four" voting scenarios in uncommon and extremely critical cases where the failure of one measurement loop must not affect the operation of the process while an alarm is raised for attendance to fix the fault.

<...Does this apply to all other transmitters...?...> Each application is different, however the principles outlined above will suffice for most applications.

Temperature transmitters always have an external sensor and always have a sensor fault condition, which is either upscale offscale or downscale offscale.
The European NAMUR association has a published standard, NAMU 43, for mA fault levels (see below).

Most industrial pressure and temperature transmitters allow one to select either manufacturer or NAMUR fault levels. The table below shows manufacturer fault levels for some pressure transmitters.

It takes some configuration, as others have pointed out, to use the fault levels on the receiving end (PLC/DCS/HMI) to annunciate a fault. It's my anecdotal impression that most people/plants just live with a very high or very low value.
A mA low fault condition infers that there is NOT a wiring issue, no cut cable/conductor/wire (a zero mA condition). The circuit has to be complete to signal the low mA condition.
Short circuits are another story - short between what and what else? Shorting the transmitter out of the loop where 24Vdc power is directly across the load resistance (no current regulation by the transmitter) is more than likely to burn out the load resistance.