Did Abraham write the Torah

Answer 1

It was Moses who wrote the complete Torah, shortly before he died (Deuteronomy 31).

Before Moses, they had Israelite parchments on which their earliest traditions were recorded (midrash Shemot Rabbah 5:18). These included the entire book of Genesis (ibid. 5:22), which had been prophetically written by the Forefathers themselves, including Abraham. The significant portions of the early parchments were incorporated into the Torah by Moses at God's dictation.

Answer 2

Abraham (18th century BCE) came from ancestry that had been God-fearing a couple of centuries earlier but had by his time slipped into idolatry (Joshua 24:2).
The Kuzari (Rabbi Judah HaLevi, 1075-1141) states that Abraham was gifted with high intelligence; and, as Maimonides (1135-1204) describes, Abraham didn't blindly accept the ubiquitous idolatry. The whole populace had been duped, but the young Abraham contemplated the matter relentlessly, finally arriving at the conclusion that there is One God and that this should be taught to others as well. This is what is meant by his "calling out in the name of the Lord" (Genesis ch.12). As a young man, he remonstrated with passersby in public, demonstrating to them the falsehood of their idols; and Jewish tradition tells how he was threatened and endangered by Nimrod.
Subsequently, Terah relocated to Harran; and it is here that Abraham began to develop a circle of disciples (Rashi commentary, on Genesis 12:5).
Later, God told Abraham in prophecy to move to the Holy Land, which is where Abraham raised his family.
He continued his contemplations, eventually arriving by his own efforts at the attitudes and forms of behavior which God later incorporated into the Torah given to Moses.
Abraham strove to raise a family (Genesis ch.15, 17, and 24) which would serve God (Genesis 18:19); and God eventually blessed his efforts, granting him many children (ibid., ch.16, 21 and 25), as He had promised (Genesis ch.17). Abraham also taught many disciples (Talmud, Yoma 28b). He saved the population of the south of Canaan from invading foreign kings (Genesis 14); and he was feared by neighboring kings (ibid., ch.12 and 20). Abraham gave tithes (Genesis ch.14), entered into a covenant with God (Genesis ch.15 and 17), welcomed guests into his home (Genesis ch.18) unlike the inhospitable Sodomites (Genesis ch.19), prayed for people (Genesis ch.18), rebuked others when necessary (Genesis ch.20), eulogized and buried the deceased (Genesis ch.23), and fulfilled God's will unquestioningly (Genesis ch.22). He became renowned as a prince of God (Genesis 23:6).
All of these forms of behavior were based upon the ways of God, which Abraham comprehended through his contemplations. These, and similar personality traits, were the teachings of Abraham and his descendants.
See also: How_did_polytheism_develop