The dove is referred to as a 'descending dove' and has biblical reference.
In Luke 3:21-22 is written (in some versions):
"After all the people were baptized, Jesus was baptized. As he was praying, the sky opened up and the Holy Spirit, like a dove descending, came down on him. And along with the Spirit, a voice: “You are my Son, chosen and marked by my love, pride of my life.”
The cross has an additional, and smaller, crossbar. As such, it is called a patriarchal cross.
That crossbar usually has the inscription INRI. (Jesus of Nazareth, King of the Jews).
Don in SoCal