Q: Is there a power-up sequence that I should follow in operating the AD7606?
A: Page 11 of the AD7606 datasheet shows that VDRIVE is dependent on the AVCC so it is worth noting that the AVCC must be powered up before the VDRIVE.
To answer your questions:
1) Roughly 5 mA is attributed to Vdrive current. But this will vary with capacitive loads and the rate of change on the Dout pins. So when designing for Vdrive you should guard band the 5 mA figure to cater for when the ADC is driving out a case where each alternative bit is swapped.
2) It is not recommended to have analog inputs when the part is not powered up. The overvoltage clamp protection is recommended for use in transient overvoltage conditions and should not remain active for extended periods. Leaving it active for extended periods may degrade the part’s bipolar zero code error performance.