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The images are not getting rendered even if the plugin is activated ?

Make sure you have inserted the shortcode properly. Start troubleshooting by inserting the most basic shortcode i.e. [twitpic id=hash]. If nothing works check if http://twitpic.com is not down.

Can I put my URL across the image ?

wp-twitpic has an option 'myurl' which overrides the twitpic.com url. But according to the twipic API notes, you need to backlink the image to twitpic page or atleast have an information around the image that the image is hosted on twitpic. So I would advice you if you override the image link with your URL please do not forget to add the credits.