To answer your first question, then yes (provided both of them are running the same software version). The software is slightly different but the API commands remain the same. Unfortuately, the AMX requires Integrator Codecs like C40/C60/C90. C20 is not an integrator, hence you will not be able to use AMX with it.
For more information on Audio Video Integration, you should check this section:
I would like to correct this reply. You can absolutely use Crestron or AMX with an SX20 or a C20. Yes, you can.
The C20 and SX20 APIs are both available on cisco.com/go/telepresence
The C20 has an optional Y cable to allow serial control from a control system.
The SX20 prefers SSH or telnet to control the system, but you can use a USB to Serial converter from the SX20. This is documented in the Synch section under Peripherals on cisco.com/go/telepresence. There is currently a bug that is being fixed in TC6.2.0 regarding use of a control system with a USB to Serial converter. TC6.2 is due for release either June or July.
That being said, our preferred platform for integrating with audio visual systems is the Integrator line, not the Quick Set line. For instance, on a C40 or better you can turn the echo canceler on or off - you cannot do this with a C20 or SX20. In addition, you can set audio recording outputs on a C40 or better. You cannot do this with a C20 or SX20.