I'm not aware of any statistics on happiness per se, most times what is represented is that there are studies showing better health outcomes for the religious than the non-religious. On the other side of the coin, excluding former Soviet bloc countries, on most measures of well-being, people in the more secular countries do better than those in less secular countries. But the research is mixed. The best article I've seen on it was a review by Phil Zuckerman a number of years back.
The part of the brain being referred to is typically called "the God center," and as with most things, is more myth than reality. From the way V.S. Ramachandran describes it, it's more that there are areas of the brain associated with the feelings involved in mystical experiences, not so much specifically relating to God. (Two of them, in fact, in the temporal lobes, iirc. [bilateral symmetry])