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Most of the fragments are from the Pentateuch, and since they follow the Hebrew text very closely, they are evidence that the first five books may very well have been authoritatively translated

That seems to be one of those truisms many people repeat (like almah being mentioned only twice), but they are simply echoing someone else who hasn't read both source.

Here is a comparison from Genesis 4, v 7 (in Tanach it's verses 6 and 7).

⁷Have you not sinned if you have brought it rightly, but not rightly divided it? Be still, to you shall be his submission, and you shall rule over him.” [LXX]

⁶ And the Lord said to Cain, "Why are you annoyed, and why has your countenance fallen? ⁷ Is it not so that if you improve, it will be forgiven you? If you do not improve, however, at the entrance, sin is lying, and to you is its longing, but you can rule over it." [Tanach]

You are right. They are way off. Is that LXX reading from the official Septuagint translation or is it from a papyrus fragment of Genesis 4? It would be interesting to find a papyrus fragment of the same and compare all three ---

BTW that TANACH translation sounds like it was done by the NIV translators. Here is a better one from Mechon-Mamre.org:

" 5 but unto Cain and to his offering He had not respect. And Cain was very wroth, and his countenance fell. 6 And the LORD said unto Cain: 'Why art thou wroth? and why is thy countenance fallen? 7 If thou doest well, shall it not be lifted up? and if thou doest not well, sin coucheth at the door; and unto thee is its desire, but thou mayest rule over it.'"