Is there a way to disable image or thumbnail link URL, as well as remove existing image/thumbnail links in functions.php?
NOT the setting under media as posted in this link, please.
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A little late to this party but it seems the OP wants to (1) prevent these WP Selfies from recurring and (2) remove existing WP Selfies.
Here is a great filter to prevent it from happening in the future: http://andrewnorcross.com/tutorials/stop-hyperlinking-images/.
And I just posted a snippet that removes existing self linking images here: http://cfxdesign.com/remove-existing-self-linking-images-in-wordpress/.
I know there is some confusion about your question, but if I understand it correctly it seems like it might be a duplicate of Disable image attachment links which I just answered with a note about a plugin that solves the functionality of removing links for images.
The plugin is called "Remove Link URL" and it removes the link by default, still allowing it to be added in if you click the button.
I believe this fixes the problem you're describing :)
If i understand right. You are looking for something like this. Key word is