Can I trigger the Yongnuo YN-600EX RT Il with the YN-622C?
The YN-600EX RT II is more or less a (more capable in some ways) clone of Canon's 600EX-RT.
It uses the Canon radio protocol.
Yongnuo developed the YN-622 radio protocol at a time before Canon had a radio protocol. At that time all of Canon's wireless flash system used optical communication which communicates via very short pulses of light.
Though both systems use the same radio frequencies in the 2.4GHz band (as do many other wireless radio devices, both camera related and not camera related), they speak different "languages" to communicate the same information.
Would I be able to do that or would I need to get a different radio, like a YN-E3-RT II? If so, how could I fit my the YN-565EX III C within that setup?
You would need an RT capable receiver, such as the Yongnuo YNE3-RX attached to the hot foot of the YN-565EX III C. The flash should be set as if it were mounted directly on a camera, as the receiver will mimic the camera's hot shoe to the flash.