Every book of The Bible mentions sin in one form or another, although the word itself is not always used.
Twelve books in the New Testament mention Abraham.
No; but there are books that mention him.
In the NIV translation as word search on the word "sin" turns up 433 hits.
I'm pretty sure that all of them mention suffering.
4
Romans
Surprisingly, the book of Esther does not mention the name of God at all.
Mathew
The books and the movies, both, don't mention it.
Yes
The Bible does not mention Jesus' date of birth or the celebration of it, therefore it does not mention "Christmas", anywhere.
There is no mention of the sin door in the Bible. Perhaps eating the sin door is an individual's allegorical interpretation of a passage in Scripture. If so, it could mean anything since we do not have any context for the question. In any case, the solution to sin for all mankind is Jesus.