Can anyone comment on the limitations and intended purpose of Processes - 'M

I believe those areas are designed to house items that will not be shared with any other users, such as the "My Models" and "My Reports" are not available to anyone else - including system administrators. You will most likely need to move your application out of those areas to share it with other users.

Aside from 'sharing' process maps and reports, can any other user run my application containing my private process model and private reports, just as though they were not private? Is there such a thing as a private Application? In other words, I can I build an Application with private parts and do full multi-user testing?

Related to this, I note Dashboard Channel Target user(s) interact with page Security settings and process map Security settings and Page share settings, such that Targetted users may or may not be able to view the channel.

Within the 'My Models' area, I don't believe anyone would be able to access to run your model - including system administrators. Same for 'My Reports'. You will need to move the model out of this area, so that permissions can be structured as needed. Outside if these areas, you can utilize process security and dashboard targeting to include/remove user permissions.

My working process is in My Models area and I set security on it, no problem. The only difference 'private area' appears to have is ability to set security on the whole 'private' folder, but this appears to be circumvented by ability to set security on individual models. Dashboards do not appear to be part of any 'private' area, and I target their channels. PROBLEM NOW is I have a big private process which totally gets screwed up when I move it outside the private area. What happens is that all process annotations disappear, all swim lanes disappear, and all node connections change from being rectilinear lines to being straight diagonal lines with no right angles. Our team has tried many various ways of Copy To..., import/export, etc. to no avail. Even after I take the screwed-up model and add the stripped annotations and swim lanes, redrawing the connections does not restore appearance or function. We are using Appian v7.6...

Humm, what's interesting is that in the private area, the View Documentation in the Designer shows the screwed-up process model, but in the Process Modeler, menu item Tools, item Generate Documentation, correctly renders the documentation. Thus the issue may not be related to copying the model outside the private area.

We have seen that before, but the most recent situations on Forum look like the pertain to CDTs being deleted. However you should be able to search for "ProcessDesignServiceXmlImpl" in the application-server.log file for more information, if that is not the case here. We used to see that on Appian 6.x before having any CDTs associated - the resolution then was something to do with restarting the server and maybe recreating analytics, or something of the sort, but I can't recall..

How are Analytics 'recreated'? Is recreating the Process Model from scratch a solution, or can it happen again? We've tried restarting the Server, but it had no effect. Would recreating the CDTs fix anything?

Once you open the affected process model, you will see an exception in the server.log which will help us get to the root cause. Can you reproduce the issue and provide us with the log excerpt here ?
The issue usually is CDT related, but it can happen under other different scenarios.
Note that you should never Save/Publish the Process model when it is in such a state.

Can you identify the path and name of relevent server log files? Id there any way to get logs from Appian environment without access to Windows Server file system? I don't have Windows Server login rights.

This issue has happened several times, and significantly impacts our tight deliverable schedule. What is the issue with Save/Publish of the affected model? I can re-add the swim lanes that disappeared, and re-draw all the annotations, and the diagonal connections are more difficult to read but seem to function. There are also other models that look perfectly normal, with only the View Documentation showing diagonal process map connections, etc. Does the 'never Save/Publish' caution apply to them as well?

You would have to work with someone who has server access. Name of the file is server.log. If you have Jboss then the server.log would be present under <Jboss_home>/standalone/log.
If the models are not having an issue then it should be fine to publish them. You should not publish a model if it is experiencing issues like these.