First off to address the latter part of your question the whole Nazism aspect doesn't really factor into Covenant. At most you could potentially stretch the concept of the Weyland Corporation and later the Weyland-Yutani Corporation being somewhat similar to Nazism due to their ruthless pursuit of the Xenomorph lifeform coinciding with their lack of interest in preserving the lives of humans in pursuit of that goal however it is a very thin parallel. The only other real correlation that could be drawn is that at the very least Weyland Corp is run by a fanatical dictator in the form of Peter Weyland and Covenant takes this further with David being a complete fanatic obsessed with genocide that more closely parallels the Nazi ideology. David views humanity with hatred and seeks to create a perfect lifeform to exterminate what he deems inferior.
As for Wagner well the Nazi's used his music because he was German and given the supremicist ideas the Nazi's conveyed, using music composed by a German was more 'pure' to their way of thinking. Whilst there is evidence to support him having antisemitic views these views appear to be consistent with prevalent opinion in Germany during the 19th Century and although being public in his expression of these views he is also noted to have had Jewish friends. The fact of the matter however remains that Wagner died in 1883 a full 50 years before Adolf Hitler became the German Chancellor in 1933 so despite his music being appropriated by the Nazi's due to it being German made and suiting their ideological standpoint from a patriotic superiority complex view. Wagner himself was not a Nazi and his antisemitic and racist views could be argued to be a reflection of 19th Century German popular opinion particular give the extent of these views is still debated amongst historians.
Now to address the music itself and Covenant's themes in general. To start off there are two pieces that are important to the character of David and those are Wagner's 'Entry of the Gods into Valhalla' and Shelley's 'Ozymandias.'Both of which David appears to subscribe to in terms of a moral outlook or at least to parts of them though on both accounts he seems to overlook important aspects of them which he is irked by when someone else points out a flaw in his ideology. The most obvious is when he attributes 'Ozymandias' to Byron which Walter quickly corrects him on pointing out it was in fact Shelley who wrote the poem. Robert Schaaf above also pointed out in his post that David also incorrectly asks 'Mother' to play Act II of 'Das Rheingold' when in reality 'Das Rheingold' is in fact a solo act consisting of four scenes. It is the first of the four music dramas that make up 'Der Ring Des Nibelungen' and is succeeded by 'Die Walküre,' 'Siegfried' and 'Götterdämmerung.'
Furthermore upon going back to 'Ozymandias' David frequently reflects on and uses the quote 'Look upon my works ye mighty and despair' when remembering how he decimated the Engineer city. David upon murdering a population with their own weapons develops a messiah complex he literally believes that he has become a God by the time the Covenant arrives at Planet 4. It is in part due to his actions of wiping out the Engineers on the planet but also because of his experiments with the nightmarish creatures that resulted from his genocide and his manipulation of them that resulted in him creating a Xenomorph that he regards as 'the perfect lifeform' that he considers himself Godlike. He appears to regard this action, when he dropped the juggernaut ships payload on the Engineer city, as being the moment he became a God and as he recognises the monstrous creatures to be indeed monsters he feels himself worthy of saying 'Look upon my works ye mighty and despair.'
Further irony is then added by Walter again who upon David expressing this statement replies, 'Nothing beside remains. Round the decay, Of that colossal wreck, boundless and bare, The lone and level sands stretch far away.' This is almost as if to comment on the death and loneliness the two see before them. David sees his genocide and manipulation of a chemical pathogen to make monsters as Godlike and something beautiful whereas Walter sees it more literally in that all David has brought is death and that in his hatred of humanity and his attempts to wipe out any memory of them he has indeed found himself alone, his perceived Empire to rule is nothing but lone and level sand stretching far away. This is almost a comment on the meaning of the poem itself that the traveler (who Walter could be a parallel of) discovers a statue of Ramesses II or Ozymandius, as he is also known, (who parallels David). The inscription on the statue reads, 'My name is Ozymandius, King of Kings. Look on my works, ye mighty, and despair.' It is an expression of power, for indeed Ramesses II ruled over the most prosperous era of the Ancient Egyptian civilisation and built a dynasty to last; just as David sees his own works. But the traveller and Walter observe that the power this great King once had is long gone, lying buried beneath the desert. In truth the poem is saying that no matter how much this powerful figure believes he has changed the world or the universe in David's case, it will soon disappear back to the dirt as with all things, a point David keenly overlooks.
Finally the significance of 'Entry of the Gods into Valhalla' actually receives a bit more explanation in a deleted scene that is available on the blu ray version of 'Alien: Covenant.' The scene is simply an extension of the interaction between Peter Weyland and David from the beginning where Weyland asks what David interprets the meaning of the song to be. David explains his interpretation of the piece to be that the Gods are abandoning their creations as they are displeased with their creations greed and vanity but as they leave to enter Valhalla they realise they too share these same flaws. It somewhat helps further understand David's character in a sense as he sees that humanity's Gods have abandoned them because they are flawed but he views the Gods as equally flawed. As he is intelligent enough to see these flaws, such as Weyland's desperation for immortality, it helps further his reasons for abandoning humanity himself. He sees himself as superior but still feels he is limited because of the fact he was created by Weyland, a human. At first he feels aligned with Weyland's goals, he wants to help his father achieve immortality and answer the unanswerable questions he seeks. However when the Prometheus arrives on LV-223 and an Engineer murders his father it seeds a deep contempt and hatred of the Engineers within him. When he destroys the population of Planet 4 he realises that he has killed the 'Gods' and so he believes that if he can create life and become a God then his creation could surely destroy the humans that he has grown to hate and view as inferior and so he begins experimenting with the black goo to the point that he manages to create a Xenomorph.
Towards the end though the use of the song I think takes on another meaning for David. He has successfully escaped Planet 4 and is now en route to Origae-6 with 2000 human colonists and embryos. He places his facehugger embryos into cryostasis with the human embryos in order to preserve them for the journey where he intends to experiment with the colonists. In this moment he is triumphant, he believes himself to be the 'Gods' and he is travelling to Valhalla, in Norse mythology Valhalla is of course the afterlife for the valiant dead, warriors and heroes who have died gloriously. In relation to David this could simply be that he is going to paradise (Origae-6), he is entering Valhalla where he will continue to perform his experiments. Though I also think it takes on another parallel in relation to the humans. David is the 'Gods,' Origae-6 is paradise, a heaven. Daniels expresses her desire to built a log cabin by a lake on Origae-6 which could be construed as a paradise and therefore Daniels is going to paradise, as are the colonists. Therefore Origae-6 is heaven. Daniels also managed to survive the events of the Covenant expedition to Origae-6 and defeat the Xenomorph aboard the Covenant, something that could qualify her as being the valiant dead which as we know, the valiant dead are the ones to go to Valhalla. So you have the Gods escorting the valiant dead to Valhalla as the narrative representation of the ending relating to the music. David is escorting Daniels and the human colonists to Origae-6.
In Norse myth the desirability of Valhalla is not simply that the classic depiction is an afterlife full of drinking, feasting, fighting and storytelling amongst the old Norse heroes but more importantly, the Gods themselves actively enter Valhalla, unlike Hel (the other Norse afterlife) A warrior dying a glorious death means they gets to go and not only swap tales with the warriors they have heard stories about and idolised as a child but they get to meet the Gods like Odin and Thor and can proudly see themselves as worthy enough to stand in their presence. The relevance of that to Covenant is that David who has a God complex views Daniels specifically as worthy. He expresses to her his intent to do to her what he did to Elizabeth Shaw, the only other human being he witnessed overcome nightmarish creatures. Elizabeth with the trilobite but also the Engineers and Daniels with the Xenomorphs. In his bizarre interpretation of things David has an admiration for Daniels, as he did with Shaw (whom he professes to have loved), primarily due to her will to survive and so he sees Daniels as the next step in perfecting his Xenomorphs. She has a value to David and is worthy of becoming a catalyst for his experiments.
The biggest influence on Covenant is 'Paradise Lost' with David mirroring Lucifer. He even directly quotes it when he asks Walter whether it is better to 'serve in heaven or reign in hell.' Like Lucifer, David intends to make war with God and create a universe with his own perversions. David sees himself as cast out by humanity who upon discovering the Engineers no longer valued his importance and he was exiled to Planet 4 which he turned into his own hellish paradise. He intends to destroy humanity for creating him so that he can establish himself as a God. Furthermore I think 'The Divine Comedies' bare some influence also, as although David is supposed to parallel the arc of Lucifer in 'Paradise Lost' if we include 'Prometheus' as well David's story is that he begins in hell during 'Prometheus' where he is a servant to the humans and isn't regarded as anything more than that, despite the value Weyland places upon him as a son but also as a financial milestone for his company.
After the 'Prometheus' expedition ends in disaster he leaves hell (LV-223) to go and discover the Engineer's homeworld though he ends up on Planet 4, which he cleanses of all non-botanical life so he can conduct his experiments using Shaw, the black goo and the Engineers themselves. It is highly probable that David discovered something aboard the Engineer ship that led to him committing genocide and developing a deep hatred of the Engineers and by extension humanity. David conducts his experiments and develops a 'perfect' organism. Though for David he feels his experiments have stalled as he desires to test them now on humans and so he lives in 'Purgatory' a state he can neither decline or progress from in his mind. Once the Covenant arrives he is presented with the opportunity to experiment on humans and successfully manages to witness them fight both Neomorphs and Xenomorphs. Not content with his creations failure to dispatch Daniels but impressed by their relentlessness he decides to perfect them more by journeying with the colonists to Origae-6 to continue his experiments and thus go to 'Paradise.' Therefore he is at the final stage of his journey, he has been through hell and purgatory and he is now on his way to paradise and so he is finally entering Valhalla, in his mind as a God, hence his request to listen to 'Entry of the Gods into Valhalla' at the end of the film.