Is it normal for a baby to move in your stomach at 3 months?

Probably not. Most first time moms feel their baby moving around 20 weeks gestation. Women who have had children can sometimes feel the baby a couple weeks earlier. However, 3 months would be a bit early, and you might simply be feeling your intestines digesting food. I concur - assuming your dates are good 3 months (12 weeks) is early - probably something else and not the baby.
No. Likely bowel gas.Most first timers can feel this by 20 wks while experienced moms pick it up by 16.