Hi Laurent,
Could you share the oscilloscope screenshot of the SPI communication error, as well as a schematic?
Does your program hold /CS low? You may be getting a glitch on the SCLK. THe only way to recover, would be to toggle /CS, which resets the SPI interface.
Regarding calibration, system calibration is used to remove offset and gain errors that are caused by the analog signal conditioning circuitry. Self-calibration, on the other hand, only removes the offset and gain errors from the ADC.
It is recommended to perform a system calibration to achieve the best performance... Since you are disabling the buffer does that mean you are using an external buffer?
Regards,
Chris