The story about Jesus forgiving an adulterous women is not in the original copy, but that's the version we most commonly have. So how does that make the Bible inerrant, given that many important parts were added or removed?
The belief in biblical inerrancy does not require that the copies or translations of the Bible be completely inerrant. It is generally accepted that:
(1) Copies are subject to human error
(2) Translations are subject to human interpretation
Biblical inerrancy is the belief that the original manuscripts are inerrant. Unfortunately, we do not have many original manuscripts (if we have any), so we study the copies that we have, compare them, and try to determine the text of the original manuscripts. We may not be able to do so perfectly, and areas of ambiguity are often noted in modern Bibles (John 8, Mark 16).
However, it stands to reason that we have a reasonably accurate representation of the original manuscripts, with the essentials preserved.
We do not have original manuscripts of any of the books in the New Testament. What we do have is thousands of copies in various languages found in a wide variety of places. Nearly 30 000 extant manuscripts of various parts of the bible exist.
We also have vast early church correspondence. One church would write to another and quote a verse and warn of other heretical teachings that where prevalent at the time. This is also a priceless accounts for the people who do new translations of the bible.
The point you raise about the woman caught in adultery. As far as I know that part is in the texts it is just lacking in the church correspondence, but not in the actual texts.
It has been theorized that the early church fathers thought that Jesus was somehow condoning adultery and that is why they did not mention that passages, but the textual evidence for the passage remains.
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