If that understanding is correct (please confirm?),

Yes, you're right

1. Does the Admin need to explicitly define at least one such project?

Yes

Or does ProjeQtor add one Administrative project by default?

No
You can define several adminsitrative project (for instance one per year)

2. How would a person know what types of tasks are available, if an Administrative-type project does not have tasks beforehand?

Name of project and name of tasks should be explicit
And tool does not prohib communication.

3. If the same meeting is attended by multiple people, then it ought to be defined by the person who called it.

If you want to folow up each meeting separately, then use Meeting object, and not on administrative project.
Otherwise, a single activity for all meetings is enough.

But usually the seniors lack this discipline: They call meetings at a whim, and do not bother to create a "meeting" task in ProjeQtor.

Laziness is the mother of all vices

One viable solution is that one of the attendees enters the task post facto, and alerts the others to enter their own timing against it.
Is that possible? In other words, if a resource creates a task, is it visible to others, and can they add their own time to it?

In this case use one single activity for all meetings (if it is just to book time)

I tried to use the Administrative project to record non-productive activities such as leaves, meetings, training sessions etc.

As discussed, I defined a project named "Timekeeping", and changed its type to Administrative.

I have not defined anything else (no allocation to this project, no assignments to these activities, etc.)

I expected that the project would become visible to our entire organization.
But the other resources cannot see the project.

Even I can see the project only in the Top+lower halves of the Projects screen.
But it is completely missing from the Planning screen (it is not even listed in the top half!).

In the Planning Screen, the project is not even visible in the top half, but I can enter activities like Leaves, Training, etc.

But after I save the activities, they disappear from the bottom half of the screen also!
This is completely freaky. Please help solve the issue.

Administrative project are not "planable" as they are not real project (they don't have a start and end date)
So they are not displayed on Gantt chart. Never.

But every resource can see activities on Administrative project on Real Wrok Allocation screen.

Well, I checked the Real work allocation sheet for a random resource (see below).

It does not show the Admin project activities at all.

First of all, I have no way to see what activities I have defined in the Timekeeping project!
As soon as enter the activity, it simply disappears, as shown in an earlier screenshot.
That's not normal!

Well, I checked the Real work allocation sheet for a random resource (see below).
It does not show the Admin project activities at all.

On the screen, parameter "hide not handled items" is not visible.
It means this parameter has been set to Yes in Global Parameter.
Then you need to move activities to status "in progress" to see them here.

First of all, I have no way to see what activities I have defined in the Timekeeping project!

Go to Activity screen.
Then select Timekeeping project on project selector

@ parameter "hide not handled items"
I tried to find this parameter in the Global parameters, but none of the 8 tabs have this parameter.
Where will I find this control?

@finding the elements in a Admin project:
Yes, I switched to Work > Activities screen, and set the project filter to "TimeKeeping".

The elements are indeed listed there.

*****************
If a resource logs in, he cannot see the Timekeeping elements on his Real Work Allocation screen:

Instead, they appear on the Today screen, in the tasks assigned to me list.

But he cannot make any use of the list, because it is a read-only list!

IMHO this is a [BUG].

I discovered another bug:

I had created the Admin-type project two days ago. Now I am seeing some spurious entries in it:

As can be seen, the meeting task now shows some resources as "assigned" to this activity.
But both assigned and left are 0 hours in all cases.

Other elements in the project do not have identical list (it varies from element to element).
But in all cases, the assigned and left are 0 hours in all cases.

Now how to interpret this display?

When I contacted these resources, they had no idea about this (they had not done anything).

Also, how to get significant MIS from ProjeQtor regarding the non-productive work being done by the resources?

There is an automatims that automatically assigns the responsible (with 0 work if (validated - assigned) <= 0
These lines should have been generated this way

This can be disabled in global parameters

I could not find that in the "Parameters > Global parameters" menu.
But I checked in the Parameters > List of Values > Status menu, and found it:

I think the help documentation needs to be made more explicit.
I will write supplementary text+screenshots and send you separately.

Thanks!

************
This still leaves the other bugs I reported in this thread:
1. Admin projects should not add resources automatically.
2. The resources see the Timekeeping (admin) tasks in Today screen, but not in Real work allocation screen.

Kindly address them.

1. Admin projects should not add resources automatically.

Recorded as Ticket #3414

2. The resources see the Timekeeping (admin) tasks in Today screen, but not in Real work allocation screen.

I think only Responsible sees it on today screen, so I think it is side effect of above remark.
If they don't see activities on Timesheet, check :
- that type of project is admininistrative (it must have code="ADM")
- if some restriction is applied to see only handled tasks, that tasks are in status that is handled (in progress)
EDIT : analysis of post https://www.projeqtor.org/en/forum/5-ask-questions/9219-how-to-enter-backdated-tasks,-and-let-projeqtor-re-schedule-the-project/reply/21036 confirms this