You will never have TTL or remote power control from a YN-622 with the YN-560III. The YN-560III is a manual-only flash. Note that there is only a single pin on the foot of the flash. This is for the "sync" (fire) signal. There are no other pins. And TTL/HSS/remote power control from the camera all happen on those other pins. The only unit that can remotely control the power on the flash is the YN-560-TX. That's it.
A YN-560-TX can tell a YN-622C-TX to fire, but not really the other way around. Basically, the 602/603/605 system is not really compatible with the 622 system. Which is why Phottix is possibly a better path to building a mixed manual/TTL triggering system. :)
Secondly, no YN-622N unit can do eTTL for Canon. The N stands for "Nikon"--it will do iTTL for Nikon. You need YN-622C units, if you want to control a for-Canon YN-568EX's power/zoom, and TTL/HSS remotely from a Canon camera. The YN-568EX is not brand-agnostic, and you should get the version that matches your camera.
You can "stack" the RF-602TX on top of a YN-622C to "relay" the sync signal if you want, but it's simpler to get additional YN-622 units. You may also want to look at the YN-685, which is apparently similar to a YN-568EX, but with a built-in YN-622C receiver.