Not sure if we can restart processes gracefully.

Yes, only works for installations.

Not sure if we can restart processes gracefully.

I think we discussed this some time ago (don't remember where and when). Restarting gracefully would at least include to reopen the repos I had opened (might be more than just 1 instance of GE), potentially also restore the window positions and other stuff.
Some internal states (selected branch/commit, filters, whatever) would probably be lost anyhow. Therefore I wouldn't bother with trying to restore something that would likely be incomplete.

This requires a full session save function, if there is none.

GitExtensions.exe --restore-last-session

By process number in the order of the launches.

LIFO

Open 1, 2, 3;
Close 2, 1, 3.
Reopen 3, 1, 2.

Like a stack. Looks like this.
Updating the top of the stack when closing.

Use case 1: Set Automatically close Git Extensions when installing plugins = FALSE.
The PM shows the dialog

Set Automatically close Git Extensions when installing plugins = TRUE
No dialog is shown

It is correct?

I think it works.

Use case 1: Set Automatically close Git Extensions when installing plugins = FALSE.
The PM shows the dialog

Set Automatically close Git Extensions when installing plugins = TRUE
No dialog is shown

It is correct?

I'm not sure I understood you correctly. Is your doubt about when the close dialog should be shown and when not?

When Set Automatically close Git Extensions when installing plugins is not checked, I expect GE to not close automatically. Instead, it should ask me (i.e. showing the dialog):

When Set Automatically close Git Extensions when installing plugins is checked, I expect GE to close automatically. Hence, the dialog is not needed, since I have nothing to choose/confirm.

Yes, only works for installations.

Maybe we should not focus primarily on the install part. IMHO, uninstall is more critical, since it can easily fail.
Just open 2 instances of GE, run a plugin in one instance and try to uninstall the same plugin from the second instance. This will fail, since the DLL cannot be deleted while it is used.
That's why all instances need to be closed before modifying the plugins via Plugin Manager.

I was confused by "Confirmation".
If this is a confirmation, then nothing should happen after the cancellation, but it is not.
Because closing GE instances is a necessity.

Only confused by the behavior after canceling the confirmation.

behavior after canceling the confirmation.

All GE instances remain open, plugin is not installed/updated/removed.