Does having a tongue ring cause bad breath?

Probably very little. Most people's tongues are cleaned somewhat by food and drinking, so that thick odoriferous bacteria does not build up. There are tools sold to scrape and clean the tongue if you are concerned, or you could just use a dull dinner type knife to scrape the tongue. What I would be worried about is breaking and cracking of teeth. I have seen teeth lost from clenching on the barbell.
Possible but........ Only if it accumulates plaque and calculus or you have an infection. If you have bad breath, it's more than likely other factors. Do your brush, floss and use a tongue scraper often enough? See your own dentist for an evaluation.