Yes, you can disable the ARGB LEDs using Corsair's iCUE software.
You can disable the lighting by opening iCUE, selecting the device and then creating a 'Static Colour' lighting layer with the RGB values set to 0 (000000). If your device supports hardware profiles, you can save these settings so that the lighting remains disabled even when iCUE is not running.
The product does not include an additional controller, as the twenty RS120 ARGBs can be controlled directly via the motherboard, thanks to precise PWM control.
The fans, as they are daisy chain on this cooler, they'll be controlled by a single header of the motherboard.
You can set up a custom curve for the fans from your computer's BIOS or a software like Fan Control.
The product "NAUTILUS 360 ARGB White" is not compatible with iCUE LINK, as it does not support iCUE LINK technology. However, it can be controlled via the motherboard's ARGB light management software. For more advanced control via iCUE, you could consider using a controller such as the COMMANDER CORE XT, but this would require an additional adapter.
The Corsair CPU liquid cooler is compatible with the NZXT H6 Air Flow RGB enclosure, as this enclosure supports heat sinks up to 360 mm in length, which corresponds to the size of the heat sink on the Corsair cooler.