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Do I need to register a new account somewhere to integrate this button?

No! Just insert the shortcode with the proper values and you're done!

What about my visitors? Will they need to register somewhere?

If they have a Facebook account (and it's more than likely that they will) then all that they will need to do is authorise the app the first time the click on the button. That's all!

Do I absolutely need an image url in the shortcode parameters?

Yes, for now the image url is required. This is the image that the visitor will see in The Want App's overlay.