First, gout is a very uncommon disease that occurs almost entirely in men above 40 years of age and very rarely in women and ONLY after menopause. You are too young and you are a woman. Therefore, the question of gout should not arise in your case.
Secondly, no gout does not affect teeth; it is only a disease of some of the joints in the feet that occurs very acutely at night after a bout of heavy drinking and big feast including seafood and red meat.