Despite finding confusing references to the 'opposites' of demons and devils and a tremulous reference to Heaven in the DMC3 manga (which could be interpreted simply as 'the supernatural world' rather than a 'Heaven' and an opposite of Hell) - which we established varies in meaning in different English translations of the Japanese - I would say that no, in the DMCverse there is no God, there are no 'angels' and there is no 'Heaven' (as Judeo-Christianity would recognise them).
Does that mean you think there are analogous concepts, i.e. Heaven or angels or God as recognized by other religious traditions?
Instead I would say that there's only the 'supernatural world' which is also the Underworld at the point in which Dante's story begins; the 'demons' are simply supernatural creatures, some of which are loyal to Mundus and some which are not, and this supernatural power can be utilised by or passed on to humans by blood as Sparda and Sanctus demonstrate. Using the word 'demon' to describe these beings is semantics, although they appear to have some typical qualities associated with Judeo-Christian demon figures; at the same time I'd say the hierarchy of DMC's Underworld is different and more based around a worship of personal strength and will to power rather than worship of Evil. The greater demons of DMC are more like Japanese Gods or figures of mythology to me than Judeo-Christian ones, less orderly and subject to more humanesque motivations.
I think the demons are obviously inspired by Judeo-Christian conceptions of them, but even among well known grimoires, the hierarchy differs. And of course, in the religious context, demons are after our eternal souls for religious reasons, rather than able to actually physically harm us. So I would say that the demons in DMC represent an even further literalization of "evil" than the notion of regular demons do. And as for their worship of power over evil, I certainly do think a case can be made that their initial sin is one of sadism, the joy in exploting and abusing those weaker than oneself, rather than pride, since there is no God to rebel against. But still, I think it is a moot point to separate evil and worship of self power (this would be interesting, the interplay between self-power and other-power, in the lives of Vergil and Sparda). In the DMC universe, the state of affairs of demons just is evil, by nature.
What human motivations do you think the demons exhibit that religious demons don't? The desire to rule? I'd say that they are less in accord with one another, but Hell is meant to be chaotic isn't it? Instinctually, all demons seem to be dirven towards the same goal: kill/torture humans and break through to the human world.
My reason for thinking that are as follows.
The DMC series revolves around the idea that demonic and human sensibilities and even forms are interchangeable, not immutable.
I think the only case where this is true is Sparda, and he may be a special case. Arkham does represent the counter-example, but it seems to me like becoming a demon is much easier than becoming a human. If form and sensibilities are inerchangeable, I believe there has to be some essence that is not, since demons are of darkness and humans are of light. But I don't disagree.
While the strong visual refs to Judeo-Christian lore and demonology in the series seem to contradict this, almost everything else goes against it. Mundus, if he were truly the "Prince of Darkness" (Satan, Ahriman etc.), would not be overthrown by Sparda (his own kind) or Dante to my mind, because he would've been put in place by God himself to achieve the task of testing mankind, he could not be removed until God himself instructed. Instead, Mundus rose to the position of the Underworld's ruler himself, and Sparda exceeded him. In the manga, Sparda is referred to as a black-horned God, which couldn't be the case in a Judeo-Christian scenario, where there is only one God. There are as many 'Gods' as may arise in the DMCverse it seems.
Your understanding of Satan is sort of occult, which although perhaps more true to the situation, might not apply to DMC. Most Christians don't believe Satan was appointed by God, but that he rebelled and was cast down, and some even believe he will never return to Heaven. You are correct that Mundus is not THE Satan referred to as constantly ruling Hell and opposing God, but then again, like I mentioned, Satan is most likely merely a title either for the ruler of Hell or for the entire entity itself. It simply means the Adversary, and grimoires give different accounts who the leader actually is, from Asmodeus, to Baal, to Beelzebub, to Lucifer, Azael, or even Belial.
So, I think that even if there were a succession of rulers of Hell, it wouldn't necessarily preclude a Christian worldview. After all, we don't really know what goes on down there, and due to the nature of Hell, never will. (But if you want to try, I know a great book on exorcism and possession called Hostage to the Devil by Fr. Malachi Martin. It is very nice, and gives an interesting insight into the paradigm of demonic possession and demons' interaction with the minds of the exorcist.)
Those lines from the manga I mentioned in the convo:
Arkham: "To me he looks like an angel."
Vergil: "You'd be wrong."
Arkham: "Semantics. Were they not once one and the same?"
Vergil: "So I hear."
...I at first interpreted this as a reference to the Judeo-Christian war in Heaven and fallen angels. There's no further mention of it though in this sense in the manga, and the translation could be painting it this way, I'm not sure. It could easily be something else. It's made even more vague by the description:
I really think the official translation should be off limits, because they are injecting Judeo-Christian concepts where they aren't present in the original, possibly as a way for us "dumb" gaijin to understand. It makes sense given how we tend to project our own religious framework onto everything else, to the point where we refer to kami as gods or Shinto as a "religion."
Scanlation for posterity:
Arkham: To me they appear as angels
Vergil: Foolish...these are demons
Arkham: That is a fact. Such is their power. You did not think there would be beauty among demons as well? It seems even the humans could unconsciously see that this was a resting place.
"Once, the legends say, heaven was at war... and the earth was ripped apart over and over again. The link between heaven and earth was rendered unstable. And from their boundaries the Underworld was born and chaos spread. The marauders became known as demons, and in the ensuing slaughter, many human lives were lost."
There it's possibly suggesting that that heaven was the Underworld, or became the Underworld (remember: "2 milleniums ago there was a war, between the Human World and the other, the Underworld" - implying only two worlds, not earth, Heaven and Hell). And it suggests that the demons were called so because they were marauders, and implies that there were other beings that were not marauders and therefore wouldn't be classified as demons.
Again, I think they are reading into things...and then writing into things. But I think the reference to marauders just means that demons were marauders compared to humans...or something. The scanlation is totally different:
"At one time one could easily return to Heaven, and countless great rents were torn in the Earth. During the birth of the Earth and the Heavens, in the instability that followed, countless times pandemonium [sic] spilled across the border. The milling humans could do nothing, and whenever the wrath of Heaven poured down on them many lives were lost."
It seems more to me that The Underworld is much like this world, there is bad in it, and good in it, and deception in it, and loyalty and betrayal, and a ruler, and all the rest.
Is it? Or are we projecting onto it? I don't think there is much good in it, or rather if there is, it has no way to manifest itself. I see the Underworld as a feudal place, with a ruling system similar to Rome or Greece, but all of that is veneer on the untameable wilderness which is Hell. But then again, demons would say the same about the human world, except we actually believe in the facades we make while they recognize them as false. So maybe you are right.
It seems natural for one to assume there would be a Heaven, but the series itself avoids it, even in DMC1's Hell, there is an appearance of a Heavenly temple in which Mundus sits (could be aesthetic deception on his part, or it could be beauty and glory depending on one's view).
I think this implies that even Mundus knows of the concepts of Heaven and angels, since he could be using it for deception, or to glorify himself, or in a manner similar to usual demonologies, mocking the authority of Heaven. Still, it would seem as if not even Mundus knows if God exists or not. I think that this is a key feature of DMC (as it is in many animes playing with Western religion), the Absent God, and the need to fill that perceived vacuum with power. This is the recognition which drives the "ungodly messiah" (cartoonish evil masterminds, or Vergil, for instance) to neurotically and irrational seek control and stability in a world that God has forgotten.
I get the feeling that this Underworld could be Heaven or Hell depending on who is ruling it - that the line between good and evil is often blurred in DMC, that the appearance of 'Hell' is sometime Hellish and sometimes Heavenly, that demons can become angels and vice versa, or humans become demons and vice versa, that created evil beings can become good, and that nothing can be taken at face value, nothing is what it seems. We play from a human viewpoint, so if the Underworld is crammed with greedy, powerful, vengeful beings then it's only natural we would think if it as a fearful place, a 'Hell'.
Do you think it is actualyl possible for the Underworld to become a good place though? Even if the ruler themself is good, why should we believe that this ideal will prevail rather than having its throat slit to maintain the status quo? I don't think the Underworld is nearly that mutable, and I believe for things to remain in balance, it must remain evil in some sense. It is unnatural for demons to act human and vice versa.