Not necessarily. "To the right" implies some kind of movement, physical or mental. So you can say: go/look to the right. "Towards" emphasizes this meaning of "to".
"On the right" means on your right hand side, and doesn't have to imply motion or direction. In everyday usage the lines are sometimes blurred.
Is there a difference between "fica à direita" and "fica na direita?" Is either one more common in Brazil?
I'd use the first option. "Fica na direita" seems informal and not grammatically correct.