Great question! The first, and probably the most important aspect in the interpretation and history of the Pentateuch (Five Books), commonly known as the Old Testament, is to understand the important Judeo-Christian traditions-- both orally and written-- that helped to shape and form the Old and New Testaments. To understand this, we have to go back to when the oral traditions were actually put onto the Sefer Torah. Most Jewish scholars would say that the very first written accounts of the Old Testament: mainly the Pentateuch of Genisis, Exodus, Leviticus, Numbers, and Deuteronomy, were put into writing by Moses himself when given the covenants by God, but most historical scholars argue that the Pentateuch were authored by several scribes through several centuries. As Martin Jaffee stated in his encyclopedia of the Torah, "The early rabbinic sages, in traditions formulated from the second through the fourth centuries CE, also believed in the Mosaic origin of the canonical text. But noticing a host of chronological discrepancies and other textual problems, they debated the possibilities that the original revelations were received and transcribed by Moses in an order that is no longer preserved in the canonical version (Mekh.Yish., to Ex. 15:9)...The fundamental assumption of modern scholarship is that the Torah, like all texts, is an essentially human product composed by authors immersed in specific historical situations. Proper interpretation of the Torah, then, involves critical study of the text, reading it for literary, stylistic, and linguistic clues to the historical setting in which it was composed (Jaffee 9230-9241)." Therefore, the reason that scribes did not cut out various parts of the early Torah is their idea that God revealed Himself to Moses wherein Moses wrote down the covenants.

As for your second question regarding discrepancies between the Old Testament and the New Testament, that can be understood as a mere difference between sects of Judaism. The New Testament continues Jewish traditions and thought, but differ in the idea of Jesus' divinity some parts the teachings of Christ. In other words, the New Testament was written in a different time period and with a different objective than the Old Testament.

References

Jaffee, Martin S. "Torah." Encyclopedia of Religion, edited by Lindsay Jones, 2nd ed., vol. 13, Macmillan Reference USA, 2005, pp. 9230-9241. World History in Context, http://0-link.galegroup.com.sable.jefferson.lib.co.us/apps/doc/CX3424503139/WHIC?u=lsc&xid=1cd35c84. Accessed 10 Dec. 2017.