It stands to reason that God is going to teach the same thing to one peoples as He does to another. He's certainly not going to one group and teach one thing, then go to another people and teach something totally different. So, it just makes sense tat a lot of things taught to the Bible people would ALSO be taught to the Book of Mormon people.
God did help Joseph Smith translate the Book of Mormon. Why would God tell Joseph Smith that the translation is exactly the opposite of what the Bible says? Isn't the gospel the same everywhere?
Also, we do NOT believe the Book of Mormon to be perfect. Joseph Smith stated that it was the most CORRECT book, but never said it was perfect. And as far as being "the most correct", that means the message, not the spelling, or the grammer, or anything like that.
You ask "those translation errors got copied right into the Book of Mormon by Joseph Smith. So why didn't God stop him and tell him the right words?"
And if the right words would seem to contradict the Bible, you believe it would be MORE acceptable? The Gospel message in teh Bible may not be perfect, but it is good enough at this point for God. Evidently, it's not broken enough for God to fix it that much.
There is no PROOF that the Book of Mormon is true or false. Just as there is no PROOF that the Bible is true or false, or the word of God. If there was PROOF, then more people would be Christian.