Actually, "pater" would almost never be used for father, the normal Latin word for father would be "abba."
"Abba" is Aramaic.
Not if he were considered a god....and he was!
The term PETER was one of the earliest names for the pagan gods. It lasted as late as Greek and Roman times. But by that time the term also took on a widespread secular meaning. It came generally to mean "father" or "parent." But this was not its primary meaning at all. The word PATER, when used in the religious addresses of the Greeks and Romans, meant NOT, as is supposed, a father or parent; but related to the divine influence of the Deity, called by the people of the East, PATOR.