Trinity

Joseph Smith taught, "Many men say there is one God; the Father, the Son, and the Holy Ghost are only one God! I say that is a strange God anyhow - three in one, and one in three!... He would be a wonderfully big God - he would be a giant or a monster" (Joseph Smith - History of the Church of Latter-day Saints, volume 6, p. 476).

"I have always declared God to be a distinct personage, Jesus Christ a separate and distinct personage from God the Father, and that the Holy Ghost was a distinct personage and a Spirit: and these three constitute three distinct personages and three Gods" (Teachings of Presidents of the Church - Joseph Smith, 2008, pp. 41-42).

But the Mormon scriptures teach Father, Son, and Holy Ghost are one God.

Doctrine and Covenants 20:28 - "Which Father, Son, and Holy Ghost are one God, infinite and eternal, without end. Amen".   2 Nephi 31:21 - "And now, behold, this is the doctrine of Christ, and the only and true doctrine of the Father, and of the Son, and of the Holy Ghost, which is one God, without end. Amen".  Alma (11:44) - "Christ the Son, and God the Father, and the Holy Spirit, which is one Eternal God".

Between 1835 and 1921, the LDS Church included the "Lectures on Faith" in its canonized version of Doctrine and Covenants.  It taught the Godhead consisted of only two Personages. See this article. Also online here or here.

Question 3: How many personages are there in the Godhead?
Two: the Father and the Son. (5:1).

Question 4: How do you prove that there are two personages in the Godhead?
By the Scriptures. Genesis 1:26: (Also 2:6).

From my research, today's Mormon missionaries believe in 3 Gods (well, 4 if you include Heavenly Mother) 
and 1 Godhead. Historical Christianity believes in 1 God and 1 Godhead.