Does drinking a lot of soda cause belly fat?

Yes. Each 12oz can of soda has 180 calories. Even diet soda causes belly fat. The artificial sweetners trigger Insulin release. This leads to fat storage.
Yes. Drinking or eating a lot of anything causes one to gain weight. The location of that weight gain is genetically predetermined. Soda does not "go to your gut" any more than ice cream or a bagel and exercises of your gut do not specifically reduce weight in your gut.