Why Mujahid can't be the author of this book
First of all none in the Muslim history attributed a book to an author in case that there are more than 1 or two intermediates between the person writing it and the attributed author. Since in general either the author himself wrote the book, or one of his students wrote it either by hearing it from the author or by being dictated it, it could happen that a close person to the student or another student completed the writing for whatever reason (see for example in your earlier question: Who wrote the book "al umm al Shafi'i") . For example in case of the book that is generally referenced to as al-Muwatta' or al-Muwatta narration of Yahya ibn Yahya al-Laithi you may find that almost each hadith or narration starts by either:
Malik informed me ... حدثني مالك
In this case we can strongly assume that the person who wrote the narration was Yahya ibn Yaha al-Laithi.
or
He informed me from Malik ... حدثني عن مالك
In this case it could be either Yahya narrating from a not named intermediate (which in one chapter should be Shabtoon also known as Ziyad ibn 'Abarrahmaan al-Llakhmi أبو عبد الله زياد بن عبد الرحمن اللخمي) or his son referring to his fathers narration from Malik
or
Yahya informed me from Malik ... حدثني يحيى عن مالك
In this case it is most likely his Yahya's son narrating via his father from Malik.
Let's get back to our book in question and explain why it can't be a book of Mujahid himself! It rather seems that this book has gone lost or scattered somehow and being gathered again later as one of the first pages show (See here), else how could one explain the mentioning of reading one of the earliest reports in Rajab 538 a.H.?
Since Mujahid died around 104 a.H. and as you claimed you may find up to 5 intermediates. On the other hand for example Yahya ibn Yahya al-Laithi died 234 a.H. and the longest chain of hadith he reports in his book reaches 4 intermediates between him and the prophet (). And since we know further that Yahya ibn Yahya al-Laithi was rather a faqih than a muhadith so he wasn't actively seeking hadith after the death iof Malik ~179 a.H.. We can strongly assume that the real author of "tafsir Mujahid" (who was a tabi'y) should have lived much later.
The chain that transmitted a huge part of this book
Note that many sources of information here are theses and papers which are generally not accessible, so one often needs to rely on second hand information. So far I only know of two editions which were verified by reviewers and commented and referenced to a certain level, the second reviewer actually intended to accomplish the job of the first.
Further scholars who have checked the book found out that there are actually narrations that are not even attributed to Mujahid himself (I have picked some examples and shared them later).
Rather the following part of the chain of narration seems the most common:
أنا عَبْدُ الرَّحْمَنِ، قَالَ: نا إِبْرَاهِيمُ، قَالَ: نا آدَمُ، قَالَ: نا وَرْقَاءُ
We were informed by 'Abdurrahman from Ibraheem from Aadam from Warqaa'
As for 'Abdurrahman he was identified as: Abu al-Qassim 'Abdurrahman ibn Ahmad ibn Muhammad al-Hamadani أبي القاسم عبدالرحمن بن الحسن بن أحمد بن محمد الهمداني who is regarded as weak and was among the teachers of al-Hakim (from Nishapur). His studies with Ibraheem ibn Disil are doubted by scholars as expressed in Siyar 'Alaam an-Nubalaa' (See here). The hadith database (see here) claims that he died 352 a.H. and he was accused of lying by some scholars. Since he was born around 270 a.H. and claimed having heard of narrations (books) of Ibraheem that the later stopped teaching around the year of 'Abdurrahman's birth and when confronted with this contradiction he stopped narrating them till his "opponents" died. So at least is rather clear that he hardly got the narrations from a face to face studying.
This would lead to the conclusion that one can hardly attribute the book to Mujahid as of sound chain nor can it authentically attributed to its potential author!
As for Ibraheem he was identified as: abu Ishaaq Ibraheem ibn al-Hussain ibn 'Ali Al-Hamadni al-Kisai أبو إسحاق ، إبراهيم بن الحسين بن علي ، الهمذاني الكسائي known as ibn Disil ابن ديزيل. A narrator which is regarded as trustful by hadith scholars (see also in Siyar 'Alaam an-Nubalaa'). And he indeed was a student of Adam ibn abi Iyas and he most likely died 281 a.H..
As for Aadam he was identified as: Adam ibn abi Iyaas آدم بن أبي إياس (died ~221 a.H.) and he is the strongest candidate for being the author of this book (see more information about him later) since he was both a muhadith with a good reputation and a mufassir. Actually many of the later shown narratives which don't mention Mujahid at all start by leaving out Aadam ibn abi Iyaas!
As for Warqaa' he is identified as: Warqaa' ibn 'Umar al-Yashkuri ورقاء بن عمر اليشكري also known as ibn Kulayb who is considered a trustfully narrator and his tafsir narrations from abu Najih are also trustful, but he didn't transmit all of ibn Najih's tafsir however. Waki' however didn't seem to trust him in matters of tafsir as stated in Siyar 'Alaam an-Nubalaa' (see here).
The author of "Tafsir Mujahid"
Among scholars it is therefore clear that the "title tafsir Mujahid" doesn't really imply that the book itself was written by Mujahid ibn Jabr, rather it is an effort of a later scholar to collect a tafsir in which only his narratives back to Mujahid are mentioned. Since I have not really read the book I must assume that this was the intention of the author.
And scholars actually named a potential auhtor the muhadith (among the students of al-Laith ibn Sa'ad and Sho'bah and the teachers of imam al-Bukhari, abu Zura'ah and ibn abi Hathim ar-Razi) and mufassir called Adam ibn abi Iyaas آدم بن أبي إياس also known as Adam ibn 'Abdurrahmen ibn Muhammad ibn Sho'ib al-Maruzi, al-'Asqalani, al-Khorasani آدم بن عبد الرحمن بن محمد بن شعيب، المروزي، العسقلاني، الخراساني. (See also in Siyar 'Alaam an-Nubalaa').
One could also assume that it is rather Ibraheem due to the explanation I've given first and this narrative where Aadam is not even mentioned. But so far available information on this book are rather rare.
Why should it even be the work of one author maybe Aadam started it and Ibraheem added some of his own "narratives" as did the son of imam Ahmad in his fathers Musnad (in this case with the result of an increase of fabricated ahadith in the book).
Further I found a narrative which is not attributed to Mujahid himself in the book here, here and here.
Some remarks on the narrators of Mujahid's "tafsir" narrations
Remark
Note that the mentioning of a "tafsir" or that a person was a "mufasir" doesn't necessarily imply that the person authored a book, rather he was known for his teaching in this topic!
In the following lines I would like to roughly name the most known students of Mujahid who transmitted his tafsir "narrations" and add a comment on the quality and authenticity of their narrations. Based on the takhrij made by a contemporary scholar (see here):
- Ibn abi Najih ابن أبي نجيح (See also here in Siyar 'Alam an-Nubalaa') he is the major narrator of Mujahid, since he had transmitted most narration of tafsir compared to any other of Mujahid's potential students. My reference claims that he has never met Mujahid and that his source/intermediate was the "book" of another student. Unlike earlier scholars my reference sees him as a man of innovation, a mudalis and doesn't trust his tafsir and he doubts whether his source was only the source I'll quote later. Ibn abi Najih was clearaly accused of being both a qadari and a mu'uatazilite nevertheless we may read that ibn al-Madini claimed that what ever was said against him his tafsir is sound. Among his students were scholars like Sho'bah (who is really a hardliner in matters of acceptance of hadith transmitters) and both Sufyan's ibn 'Uyyanah and a-Thawri (the first in fact called him mufti Mekka after 'Amr ibn Dinar).
If we accepted the view of this contemporary scholar which clearly is in opposition of earlier scholars we must assume that the whole chain of this book is of no use... but Allah knows best!
- Khossaif ibn 'Abdarrahman خصيف بن عبد الرحمن also known as abu 'Awn (See here in Siyar 'Alaam an-Nubalaa'): His narration is rare and he is rather a weak narrator whom's narratives can't stand alone as an evidence.
- Ibn Juraij ابن جريج (see also here in Syar 'Alaam an-Nubalaa'), here ibn Ma'yn calimed that his narrations from Mujahid are rather weak while ibn Hebban even claimed that he never heared Mujahid's tafsir/hadith and his source/intermediate is the same book as stated earlier for ibn abi Najih, however he learnt two ahruf from teh qira'at from Mujahd as claimed by a-Dhahabi.
- Laith ibn abi Sulaym ليث بن أبي سليم also known as ibn Zunaym (See also here in Siyar 'Alaam an-Nubalaa') his hadith and narratives from Mujahid are few and he's regraded as rather weak due to his memorization. Most of his content is present in the book of our "so far" unknown intermediate, nevertheless Yahya ibn Sa'id al_Qatan (another hard liner in matters of choosing hadith transmitters) accepted his tafsir narratives.
- Sufyan ibn 'Uyyanah سفيان بن عيينة (107 a.H. - 198 a.H.) (See also here in Siyar 'Alaam an-Nualaa') his tafsir narratives are also rare, but his trustworthiness is mainly accepted. You may notice that he shouldn't be a student of Mujahid therefore its most likely that he also had our unknown intermediate as a teacher. As a matter of fact none of the "general" sources even claimed him to be a student of Mujahid.
- Yazid ibn abi Ziyaad يزيد بن أبي زياد (47 a.H.- 137 a.H.) (See her in Siyar 'Alaam an-Nubalaa') he was a teacher of the two Sufyan's and Sho'abah our source calls him a Shi'ite and regards him as da'if and so did Ahamd and ibn Mai'yn too. A-Dhahabi also stated a statement fo ibn Fudayl calling him "one of the biggest imams of the Shi'a".
- 'Ataa' ibn as-Saa'ib عطاء بن السائب (died 136 a.H.) (See also here in Siyar 'Alaam an-Nubalaa') his reputation as narrator is good he also reports via Mujahdid from ibn 'Abbas.
- Al-Qassim ibn Bazzah القاسم بن أبي بزة (died 145 a.H.) most of the narrators who transmitted Mujahid's tafsir statements used his book as an intermediate he is highly accepted as a narrator.
Off-topic: The roots of early hadith books of Mujahid's tafsir
Basically all tafsirs rely on ibn abi Najih and none mentioned Al-Qassim ibn Bazzah that's why I've put his name as a potential intermediate in brackets (See also here and in the following pages)
- As already stated the tafsir we are about to discuss used the root Mujahid -> (Al-Qassim ibn abi Bazzah ->) ibn abi Najih -> Warqaa' ibn 'Umar al-Yashkuri ...
- Ibn abi Hathim: Mujahid -> (Al-Qassim ibn abi Bazzah ->) ibn abi Najih -> Warqaa' ibn 'Umar al-Yashkuri -> Shabaha ibn Sawwar شبابة بن سوار (trustworthy) -> Hajjaj ibn Hamzah حجاج بن حمزة (sadooq).
Also via:
Mujahid -> (Al-Qassim ibn abi Bazzah ->) ibn abi Najih -> Shibl ibn 'Abbad شبل بن عباد (trustworthy, but qadari)
- A-Tabari: Mujahid -> (Al-Qassim ibn abi Bazzah ->) ibn abi Najih -> 'Isa ibn Maymun عيسى بن ميمون (trustworthy, but qadari) -> Abu 'Assim an-Nabil أبو عاصم النبيل (trustworthy) -> Muhammad ibn 'Amr al-Bahili محمد بن عمرو الباهلي (trustworthy). This chain appears around 700 times in this tafsir and is regarded as the sahih chain of Mujahid's tafsir in this book. But also: the 2nd root mentioned earlier for ibn abi Hathim via Shibl ibn 'Abbad.
- Al-Baghawi: Mujahid -> (Al-Qassim ibn abi Bazzah ->) ibn abi Najih -> Muslim ibn Khaalid a-Zinji مسلم بن خالد الزنجي he was said to have a weak memorization ability.
