Some of those have already been answered in this thread, but just to give a complete response:
There isn't an option to start a closed conversation.
Moderators see what you saw in the video in the front end. Moderators and administrators are two different functions, administrators won't have access to the moderator tools unless they are given the relevant permissions.
The person issuing the warning is shown in the overlay.
If you don't start a conversation the recipient is unaware, unless it is a public warning.
If moderators have the Manage all warnings permission then they can delete any warning.
If members have the View warning details permission then they can view all warnings, on their own profiles and others (a suggestion has already been made to view own warnings only).
The note is displayed in the warning overlay.
All warnings are templated. You can remove the comments section from the template.
You could probably also remove the conversation section to stop it being edited.
If you want to make all warnings public, give all members the View warning details permission.