All Mormons believe this? Is it part of their doctrine
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All the mormons I knew did. It was taught at BYU, in Sunday school, Sacrament meetings, Joseph Smith taught it, as have other leaders down through the history of the Church.
If God was once man, which they also teach, then He lived on an earth, and had to live the ‘mormon’ gospel so he could ‘progess’ to godhood and had a God of his own to worship.
Here is just one quote...
“But if God the Father was not always God, but came to his present exalted position by degrees of progress as indicated in the teachings of the prophet, how has there been a God from all eternity? The answer is that there has been and there now exists an endless line of Gods, stretching back into the eternities, that had no beginning and will have no end. Their existence runs parallel with endless duration, and their dominions are as limitless as boundless space.” - B. H. Roberts, New Witnesses for God, 3 vols., 1:, p.476
I don’t know if there is a legitimate way to answer your first question truthfully, because I don’t know how to poll “All mormons” find out what they believe on the matter, I know I believed it at one time and my wife believes it now.
What I do know is that it is part of their doctrine. D&C Section 132 speaks about men becoming gods. That all mormon males are “progressing to godhood” to eventually rule over their own planet and populate said planet with spirit babies.
Here’s something from JS about this:
Joseph Smith – Prophet, President
God himself was once as we are now, and is an exalted Man, and sits enthroned in yonder heavens. That is the great secret... …I am going to tell you how God came to be God. We have imagined and supposed that God was God from all eternity. I will refute that idea, and take away the veil, so that you may see. … It is the first principle of the Gospel to know for a certainty the character of God and to know...that he was once a man like us.... Here, then, is eternal life - to know the only wise and true God; and you have got to learn how to be Gods yourselves, and to be kings and priests to God, the same as all Gods have done before you... (“King Follett Discourse,” Journal of Discourses 6:3-4, also in Teachings of the Prophet Joseph Smith, 345-346, and History of the Church, vol. 6, 305-307, emphasis added)
Hope it helps to answer your question(s).