Bald eagle pops up

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and try to change the photo there. Then reboot and check your picture..

NOTE: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

If you hover the cursor over your Pic in System Preferences, the word "Edit" doesn't appear?

If edit doesn't appear, try dragging a picture file onto the image.

On the 3 computer I Own and am the Administrator of - it works fine as intended.

Sorry, but can not duplicate this issue.

Then would suggest;

1 - Get Support Choose a product and we’ll find you the best solution.Start now and open an Apple Support Ticket as they are Apple Employees to deal will these types of issues 

Is there a padlock at the bottom of the pane?

I don't think there is, but don't understand why you cannot change your user account picture.

If you hover the cursor over your Pic in System Preferences, the word "Edit" doesn't appear?

no it does not what ?

No Edit, or something else

if something else - what does it present ?