What is the importance of the theme of life after death in the Book of Job? Actually, it is VERY important but the recognition of this question & the response in the Book are barely on the periphery of our understanding when one is focused on the popular Central Question of the Book, i.e. the meaning of suffering & evil (i.e. theodicy.) Expand that focus to see the TRUE Central Question of the Book, i.e. "What is man," & everything changes. Life after death can be imagined as a huge planet that orbits the True Central Question. When the question of suffering & evil is seen as the center of the Book then this central question all but completely blocks the view of the 'planet-of-life-after-death.' But when suffering & evil is recognized as a huge red-hot planet that orbits the True Central Question, along with others, it no longer blocks our view & the 'planet-of-life-after-death' comes into full view.
The OP question in the Title, i.e. does Job believe.., is only a PART of the larger a question that is - what does the Book of Job say about life after death? This is the more important question in which to frame the more specific answer about Job's beliefs.
To begin, in the light of the True Central Question, the importance of [hasatan's] accusation against all men is highlighted. "All that a man (any, every, & all men) has he will give for his life.. touch his bone & flesh & he will curse (God) to (His) Face." (See my answer to the Question "What did Satan mean, when he replied to God, "Skin for Skin"?"”) [hasatan] is willing to contradict the LORD's judgement of Job because [he] has been all over the world & seen what man was. (See my Answer to the Question, "Was Satan joking with God in Job 1: 6-7?")
[hasatan's] position was Job was no better than any man who would curse God to His Face under adversity.
Next, in the light of the True Central Question the words of the [spirit] that comes to Eliphaz is chapter 4 are highlighted. 4:17-21 (KJV) states:
"Shall mortal man be more just than God? Shall a man be more pure than His Maker? Behold, He put no trust in His servants; And His angels He charged with folly: How much less in them that dwell in houses of clay, Whose foundation is in the dust, Which are crushed before the moth? They are destroyed from morning to evening: They perish forever without any regarding it. Doth not their excellency which is in them go away? They die, even without wisdom."
In these words the assertion can be clearly recognized that man's death is final, man does not deserve life after death.
Next, in the light of Central Question, the poisonous vitriol in the [spirit's] statement about man can be recognized as permeating the discourses of the 3 friends (as discussed in my answer to the Question "What is Eliphaz explaining in Job 4:12-21?") This table highlights this poison.
Man is corrupt & abominable, a maggot & a worm. There is no hope for man, death will & should be his end forever.
Next, in the light of the True Central Question, Eliphaz's accusation against Job, Chap 22, "Is not THY (Job's) wickedness great.." & Bildad's accusation against 'wicked men' (Chap 18) & Zophar's accusation against 'wicked men' (Chap 20) can be recognized for what they really are, accusations against all men. Zophar's statement (20:7) is an abt summary of the 3 friends accusation against man "(man) will perish forever like his own dung." It is also an apt projection of [hasatan's] contradiction of the LORD before the heavenly assembly & the [spirit of contradiction’s] vitriol against man spoken to Eliphaz. (NOTE: it is not possible to fully support these & following statements in this limited space here. It would make the answer way too long. For proofs supporting these statements see my videos & my analysis posted on the web mentioned in my Profile.)
Standing opposed to these projections that man will die forever is Elihu. We see a response to this death sentence when Elihu begins telling Job that God (36:16) "has allured you out of distress into a broad place where there is no cramping & what was set on your table was rich food." Part of the rich food set before Job was that God "does not withdraw His eyes from the righteous, but with kings upon the throne He sets them FOREVER, & they are exalted." (36:7)
Certainly Elihu's response defends man & stands in opposition against Zophar statement that man "will perish forever like his own dung." We hear his hope for man in v. 33:23-30 when Elihu speaks of the "light of life" for man. Elihu's uplifting words about man stand opposed to the poisonous vitriol that can be recognized in the words of [hasatan], the [spirit of contradiction] & (unwittingly) in the words of the 3 friends.
Life after death is very important in the Book of Job. It is part of the response to the Central Question, 'What is man,' supporting that man can be blameless & upright, fearing God & avoiding evil, whom God can & will justly seat with kings on their thrones forever.
Within this understanding & framework concerning life after death in the Book of Job, the contradictions that the OP points to in Job's responses concerning life after death can be set within a proper framework & understanding. This understanding may be expressed using an analogy. A human author may have a character present contradictory opinions regarding an important & perplexing question in a Book. The point is to examine the different perspectives in an attempt to prepare for the revelation of the resolution to the question.
God has blessed us in the words of Job, as Job presents the implications of his excruciating sufferings & how his lack of understanding as to the reasons for this suffering can lead to loss of all hope. For many, these doubts & thoughts of Job resonate with our own feelings in the face of excruciating suffering. It can be comforting to hear the resonance of our own thoughts & concerns in Scripture. Yet the clear message of the Book stands high above the turmoil & the anguish, that God is so great that we should never call Him to court to justify Himself. After all, Who is like God?!
As part of an aside concerning this issue, 2 things should be noted.
1st - for Catholics (& Orthodox) the affirmation of life after death has already been clearly stated in the Book of Job. The official translation of the Catholic Church, the Latin Vulgate, contains specific statements regarding life after death that are not part of the Masoretic text. (These statements are already described in the other Answers to this post.) These statements together with clear statements of life after death in the Wisdom of Solomon & the Book of Maccabees (both part of the Catholic canon) (& elsewhere) makes it very clear to Catholics that the Book of Job is written with a context confirming life after death.
2nd it should be noted that there is a connection between the discussion of life after death & the existence of angels. Saint Paul noted that the Sadducees did not believe in the Resurrection, nor angel, nor spirit (Acts 23:8.) Angels are spirits & the implication of the existence of spiritual beings means that man's spirit may continue after his bodily death. The existence of angels is not positive proof that there is life after death but it certainly part of the supporting evidence used to support faith in life after death. Angels (sons of God) are mentioned many times in the Book of Job & are supporting evidence that the Book of Job, as clearly stated in the Greek LXX Septuagint, is framed within a faith in the existence of life after death.