Why is there no mention of temple rites of any kind, (baptism for the dead, eternal marriage, becoming gods) spoken of in the Book of Mormon?
The Book of Mormon was not written as a handbook of religious beliefs. The purpose of the Book of Mormon is principally to the convincing of the Jew and Gentile that JESUS is the CHRIST, the ETERNAL GOD, manifesting himself to all nations (Book of Mormon, title page).
Temples are mentioned throughout the Book of Mormon. Nephi, who came to the Promised Land from Jerusalem with his father’s family and others in 600 B.C. built a temple after their arrival, which was patterned after the temple of Solomon.
The word of God was taught in the temples–
Wherefore, I must tell you the truth according to the plainness of the word of God. For behold, as I inquired of the Lord, thus came the word unto me, saying: Jacob, get thou up into the temple on the morrow, and declare the word which I shall give thee unto this people. (Jacob 2:11) ~540 B.C.
And now, it came to pass that Mosiah went and did as his father had commanded him, and proclaimed unto all the people who were in the land of Zarahemla that thereby they might gather themselves together, to go up to the temple to hear the words which his father should speak unto them. (Mosiah 1:18) 124 B.C.
And Alma and Amulek went forth preaching repentance to the people in their temples, and in their sanctuaries, and also in their synagogues, which were built after the manner of the Jews. (Alma 16:13) 78 B.C.
And now it came to pass that there were a great multitude gathered together, of the people of Nephi, round about the temple which was in the land Bountiful; and they were marveling and wondering one with another, and were showing one to another the great and marvelous change which had taken place. (3 Ne 11:1) 34 A.D.
Temples are prominently mentioned in both the Old Testament and New Testament as sacred places of worship, yet very little is mentioned in the bible about temple ordinances. We find more in Egyptian archaeology than we do in the Bible about temple worship in the Old Testament as articles of clothing closely approximating the temple clothing of today are found on some Egyptian mummies.
As found in the Book of Abraham in the Mormon scripture, the Pearl of Great Price, the first Egyptian pharaohs, although not authorized to conduct temple worship, attempted to copy temple rites from the patriarchal fathers.
Now the first government of Egypt was established by Pharaoh, the eldest son of Egyptus, the daughter of Ham, and it was after the manner of the government of Ham, which was patriarchal.
Pharaoh, being a righteous man, established his kingdom and judged his people wisely and justly all his days, seeking earnestly to imitate that order established by the fathers in the first generations, in the days of the first patriarchal reign, even in the reign of Adam, and also of Noah, his father, who blessed him with the blessings of the earth, and with the blessings of wisdom, but cursed him as pertaining to the Priesthood. (Abr 1:25-26)
Although the word “temple” is mentioned 106 times in the New Testament, not one word is spoken about what goes on in the temples other than they were places for teaching the gospel, as exemplified in the gospels of Matthew and Mark,
In that same hour said Jesus to the multitudes, Are ye come out as against a thief with swords and staves for to take me? I sat daily with you teaching in the temple, and ye laid no hold on me. (Matt 26:55)
And Jesus answered and said, while he taught in the temple, How say the scribes that Christ is the Son of David?(Mark 12:35)
So we see that the Book of Mormon is consistent with both the Old and New Testaments in its treatment of the details of temple worship.