Some believe the saying came about because people believed your heart stopped for a second when you sneeze and your soul briefly leaves your body. The "bless you" is a shield against evil, so your soul returns.
The National Memorial Ladies was created so that no soldier would ever be buried alone. A woman noticed many funerals at, I believe, Arlington National Cemetery where she visited her husband's grave, without any family or friends. She was determined to attend those funerals. At least, that's how I remember it. I went to the site and couldn't find a history on it.
As far as "God bless you" and "Bless you," the southern baptist church I grew up in didn't believe that we lay people should bless anyone. That was left to God and the pastor, who spoke for God. And why say that? What's wrong with "Deepest sympathy?"
