The camera control function of the Atomos Shinobi II is not supported by the Canon 1DX Mark III. However, the image signal via HDMI works normally, only camera control via the monitor is not possible.
The monitor is compatible with the Sony ZV-E10, including camera control via USB-C and touch-to-focus function. No explicit compatibility confirmation is given for the Sony a6700, but as the monitor supports many Sony Alpha cameras with USB-C tethering, compatibility is likely provided the camera supports USB-C tethering and has the latest firmware.
The Atomos Shinobi II can be supplied with power in several ways, so that a separate battery is not absolutely necessary, but can be used as an option.
1. USB-C PD: The Shinobi II can be powered via the USB-C port, which is a convenient option, especially if you also want to charge your camera device via the same port.
2. NP-F battery: The monitor can also be powered by an NP-F style battery, which is a good option for longer-lasting shots.
This means that a separate battery is not absolutely necessary, but it offers flexibility and longevity in terms of power supply.