Does it run at 6,000 MHz out of the box?

Not automatically. You need to enable EXPO on AMD boards or XMP on Intel boards in the BIOS to reach the rated 6,000 MHz.

Is 32GB enough for gaming?

Yes for most games and multitasking. 32GB leaves headroom for browsers and background apps, while 64GB mainly helps heavy editing or virtual machines.

Will it fit under a large air cooler?

Check cooler clearance. The heat spreaders are taller than bare sticks, and some large tower coolers overhang the first memory slot.

Does it work with AMD AM5 and Intel LGA1700?

Yes, it is sold in kits tuned for both platforms, but confirm the exact part number is on your motherboard's memory support list.

Do I need Corsair iCUE?

Only for lighting control. The memory runs at its rated speed without iCUE, but RGB customization and sync require the software.

Is it better than a 2x16GB 5,600 MHz kit?

The higher 6,000 MHz rating gives a small bandwidth edge on current platforms. A 5,600 MHz kit is a reasonable alternative if your board or CPU memory controller is limited.