Can I trigger a YN565EX using a YN600EX-RT?
Yes, but probably not the way you're thinking. You can put the YN-565EX into S1 or S2 "dumb" optical slave modes, and use the YN-600EX-RT on-camera as a master, but this is a manual-only method of triggering, and has all the range and line-of-sight issues of optical triggering. In this system, it's a simple "sensor sees flash; fires flash" operation.
Most people say that the 565EX has no built-in receiver. But I've read the instructions and there are features to set-up groups (a,b,c) - I think if the YN565EX could be used only as an optical slave then there would be no reason for this feature. (If there is a groups set-up in the flash menu then it must be triggered by radio trigger?)
Actually, the YN-565EX can be used as a "smart" optical slave in Canon's optical wireless eTTL system. This is where the master flash and the receiver exchange flash signals, rather like Morse code, to communicate most of the same information that is done over the foot/hotshoe pins/contacts when the flash is on-camera. This is why it has group settings--for use in Canon's wireless eTTL scheme. But the YN-600EX-RT, unlike the Canon 600EX-RT, has no optical master capability (it can be used as a slave in the system, though--all the Yongnuo flashes with EX in the name can). To use this capability in the YN-565EX, you'd need an on-camera Canon wireless eTTL master unit on the camera hotshoe. But if you got one, you could control both of your flashes off camera.
But, again, you would have the range and line-of-sight restrictions of optical triggering.
I also did some research and there is no information about any built-in receiver (whether it is 2.4GHz or not)
There is no built-in radio receiver in the YN-565EX. The only Yongnuo units with built-in radio receivers are the YN-560 III/IV and newly announced YN-660 (to be used with the RF-603/605 and YN-560-TX triggers), YN-685 (to be used with the YN-622 triggers), and YN-600EX-RT (to be used with Canon's RT system). All of these triggering systems are incompatible with each other and do not, for the most part, work together.
If you want to use the YN-565EX in the Canon RT system the 600EX-RT "speaks", you'll need to add a YN-E3-RX receiver unit to its foot.