So... there are different opinions on this. Here is my take - I am sure others will violently disagree. Then I will provide a direct answer :-).
I don't use ZHO as most of our systems don't require it. If you were working with older inventory scan guns and perhaps some VOIP applications this can be a requirement, but for most of the systems the typical integrator do it is more of hindrance, I repeat MOST. Since the ZHO systems need to be on the channel you need to take care in setting up cell sizes and all the parameters that go along with that.
We did testing of ZHO in our office and a couple of deployments, it worked less reliably than a multi channel roaming design. I have gotten better results by setting the min RSSI, band steering, channel and power plan, etc... than screwing around with ZHO. I have had similar results with Package from a couple years ago - probably different now. For us - ZHO is a PITA, doesn't really work well and just does not make senses. Please don't hate on me. Again, I understand the some systems require ZHO to function - if so use Aruba or the like.
Now - to answer your question directly - to my knowledge the new AC units to not support ZHO at the moment, and if Ubiquiti is to be believed - never will. I do not have a cloud key site with old APs to look up, but I assume if you were using the original AP, AP-LR, or AP-Pro, that you would find the ZHO setting under the WLAN group area. I also think this feature has come and gone on different versions of the controller. So - short version is yes and maybe :-).
The CloudKey runs the same controller software as server regardless of where it is located. One minor disadvantage of a cloud key vs AWS or private server, is that you do not disconnect notices if an AP or entire system drops offline, i.e. probably ISP outages and problems. I understand this functionally is coming in future releases, but there are many other ways to get at that info if you have need.... I already got my ass kicked over this on... I hope he is not reading this....
Sorry for the long response to a single sentence questions!