When I enable the template hints (System->Config->[select store/website]->Developer->Debug->Enable template hints), why does Magento show the standard and well-loved 'favicon.ico' file instead of the 'favicon.ico' I have uploaded for my theme/site?
Ok so I can take an educated guess on this one.
When you upload a favicon via the admin Magento will not replace the default favicon.ico image but rather upload and new one and point to that one with some html e.g:
<link rel="icon" href="http://website.com/media/favicon/default/favicon.ico" type="image/x-icon" />
When you turn template hints magento outputs HTML some of which is not valid not valid, for example the template hint div is echoed before the doctype:
My guess is that some browsers such as Firefox will still correctly interprit the HTML (and the ne favicon link) and others such as Chrome will not be able to.
In the root of your Magento site there is a favicon.ico file if you replace this with your own (this can't be done via the magento admin) then that should be picked up even when template hints are turned on.