Will a Yongnuo YN-622-TX operate a Yongnuo RF-605 attached to a remote YN-565EX flash, and will it allow the 622 to perform eTTL operability with the remote flash?
Also will the YN-622-TX operate a YN-560IV flash, albeit in manual mode?
All three of the Yongnuo radio triggering systems are incompatible with each other.
The "manual" Yongnuo triggers (RF-60x/YN-560-TX), the TTL Yongnuo triggers (N-622/YN-622-TX), and the RT Yongnuo triggers (YN-E3-RT/YNE3-RX) have differing communication protocols, despite all being 2.4 GHz radio triggers. The only way to get them to interoperate is to stack triggers (I.e., put an RF-60x unit on the passthrough hotshoe of a YN-622), or to use the bassackwards "compatibility" mode on the -TX units (i.e., you can control YN-622s from a YN-560-TX by way of a YN-622-TX).
The YN-622 cannot (by itself) set off a RF-605. The RF-605 is a manual-only trigger and cannot communicate TTL/HSS. All it can communicate aside from the sync signal is the wake-up signal, and group on/off, if the transmitter is a YN-560-TX. And the YN-560-TX is the only Yongnuo transmitter that can control the power/zoom on a YN-560III or YN-560IV remotely. So, unless you put a YN-622 on the foot of the YN-560IV, no, the YN-622-TX cannot trigger or control a remote YN-560IV.
If you want manual and TTL triggers that are designed to interoperate together, you should probably look at another brand of radio trigger, such as Phottix, RadioPopper, or PocketWizard, who all make both manual and TTL triggers that work together.
If you want wireless TTL over radio with your YN-565EX and you want to use radio triggers, your best bet is to go with the YN-622 transceivers and a YN-622-TX. Or, if you shoot Canon, possibly getting a YN-622-TX and a YN-685 (if you want a built-in receiver). Also if you shoot Canon, and you're considering the RT system, the YNE3-RX can be used as an add-on receiver in the RT system.