This is just a quick gander into the way the relationship between Aziraphale and Crowley progresses in Good Omens throughout the book, probably in not too eloquent a format.
What strikes me is that at the beginning, it is portrayed as very much a matter of necessity/convenience rather than emotional fulfillment: "they wouldn't have chosen each other's company voluntarily" is how the book puts it, and presents the saving of time/expenses as the main, practical benefit of the Arrangement. This is obviously what the two (want to) think of the Arrangement, and somewhat in contradiction to what is actually shown: they're both intimately aware of each other's hobbies and interests and make a habit of dining or feeding the ducks together. It is clear their relationship already has an emotional level that goes beyond the necessary, but they do not actively acknowledge it.
How that comes to change is one of the running themes in the book.
The presumable starting point is Crowley's idea of collaborating on the Raise the Antichrist project. Though most of it is off-screen, it can be assumed that in that time, they come to work together more closely and meet much more frequently than they used to. (For comparison, when they first meet in the park, Crowley mentions owing to pay for lunch from the time of the French Revolution, roughly two hundred years previous)
They are in close to constant contact for much of the week before the scheduled end of the world, except for the time Aziraphale is swallowed by the engaging prophecy book and the delay after. The discovery of Adam's location is a decisive moment for the angel: He realises that while he ought to tell his superiors, what he wants to do is tell Crowley, that the Arrangement is far more than simply a way to go about his duties more efficiently, that his desires and duties are, at once, in conflict. (Note that while they have been objectively irreconcilably in conflict at least since he decided to try and avert the Apocalypse eleven years previous, he only agrees to it after Crowley persuades him to rationalise it as thwarting Hell's wiles.) This self-discovery is cemented after he goes ahead and contacts Heaven anyway in a sort of naive hope, and when Heaven wants to go ahead with destroying the world, is hit with the bitter realisation that he and Upper Management are officially at odds. He proceeds to try and contact Crowley at once but is interrupted.
This parallels Crowley's own point of development: While Aziraphale is reading the prophecy book, Crowley is sitting around in his flat killing time, trying to distract himself without much success. As the one who screwed up Hell's plan, he's in a bad spot and is waiting for the other shoe to drop. After he temporarily repels Hastur and Ligur, when Hell is literally on his heels, the first place he goes is the bookshop - not because he knew Aziraphale had found the Antichrist (he didn't know - he never heard the message the angel had left on the answerphone), but because the angel was the only person who could be counted on to help. His only friend, not just the counterpart to their "tacit agreement of non-interference", as the Arrangement is described early on. When finding a burning bookshop and a missing Aziraphale, Crowley panicks - not over Aziraphale's safety, since at worst the angel would have been discorporated, but over the prospect of being left alone in his great mess (and probably never seeing Aziraphale again).
That's a suitable moment to step back and consider the differences in how Aziraphale and Crowley approach each other. The first point is Crowley's consistent neediness, for lack of a better term, as opposed to Aziraphale's relative self-sufficiency and tendency for distraction. This starts as early as the first scene in Eden, where Crowley is the one to initially approach Aziraphale - it's not spelled out, but the angel has guard duty and has to stay in one spot while the demon moves about. Therefore, it's likely Crowley was the one who initiated first contact because he wanted to, which Aziraphale accepted more out of politeness and boredom than anything else. (Crowley, in turn, presumably picked Aziraphale to talk to because the other animals were boring, and the cherubim at the other gates too quick to smite.) Their first appearance in the book involves Crowley being completely engaged in the conversation and eager to keep the other person engaged, even as Aziraphale drifts off, distracted by the changing weather. (That should feel familiar to anyone who's ever tried to pull the weight to keep a conversation going despite the other person not being equally motivated.)
This trend continues in the present day, first with the subtle differences between their places of residence: Crowley's flat has the unlived-in look that "comes from not being lived in", implying that he doesn't spend much time in there, while Aziraphale doesn't seem to leave the bookshop much and can easily get lost in reading for days (or however long it takes for cocoa to develop green fur). In a word, the impression one gets is that Crowley prefers not to be alone, while Aziraphale really doesn't mind it. Similarly, there's a mention of how Aziraphale "banishes" evil: He mentions having work to do and the demon usually gets the hint, implying that Crowley makes a habit of dropping in and hanging around until asked to leave. This difference is shown most clearly when they part ways after meeting Anathema: Crowley repeatedly asks Aziraphale, sounding rather nervous, if they'll meet up later, if that's okay, tries to prompt him to give a reassuring answer while Aziraphale is absorbed in his discovery of the prophecy book. Then, Aziraphale finally notices him and distractedly agrees before shutting the door on him, leaving Crowley "feeling very much alone".
(I'd like to retreat from my professionalism at this point and give Crowley a big hug. There you go, Crowley.)
Thus, we come to the events already described above: Crowley feeling lonely and anxious in his flat while the angel reads, them jumping to contact each other as soon as the going gets rough, et cetera.
But there's another, very crucial point to consider before we wrap this up. In a very blunt nutshell, Aziraphale thinks Crowley is evil.
For the longest time, it is one of those intrinsic cosmic truths he doesn't seem to question much, ineffable as they are. Back in the Garden, he calls Crowley out on being a demon and only being able to do evil, and misses his sarcastic comeback to that. It's "nothing personal", but there it is.
Six thousand years later, his outlook hasn't changed as much as you'd expect. While he's cooperating with Crowley without much concern, you get the sense that while Crowley doesn't think much of dealing with an angel, Aziraphale has never forgotten that he's dealing with a demon. During their drunken dolphins conversation, he expresses concern that Crowley is trying to do his thing on him, i.e. trying to tempt him, and approaches their agreement, complete with shaking of hands, with visible if minor trepidation. In that same conversation, when they discuss the future of the Antichrist and Aziraphale brings up good and evil, Crowley calls him out on it: "They're just names for sides. We know that." It seems more accurate to say that Crowley knows it, and has been trying to get Aziraphale to understand it for the past six thousand years, as well. When Crowley describes the two of them as "occult", Aziraphale insists that demons are occult but angels are "ethereal", is reluctant to be the first to reveal his list of organisations ("But you're a demon!") until Crowley gives his word to do the same, and also dismisses Crowley's attempts to understand the love aura the angel is registering in Lower Tadfield, saying that he can't describe it better, "especially not to [Crowley]".
In short, it can be said that Aziraphale approaches Crowley with some degree of prejudice and caution, if not looking down on him then certainly considering him to be Different - perhaps not without cause, as in the cosmic scheme of things, he has, in some respects, more to lose from trusting too much: a demon cannot Rise, but an angel can Fall, and he knows at least one angel who Fell - sorry, "sauntered vaguely downwards" - from "getting in with the wrong people", which is exactly what he'd be doing by making the wrong judgement call. (By contrast, Crowley's trust in the angel is unwavering, to the point where, as the Apocalypse begins to unfold and their respective sides prepare to start slaughtering each other again, it doesn't even occur to him that their Arrangement could in any way be affected by that, and he attempts to contact the angel for help.)
The reader is treated to a more rounded view of Crowley and demons in general. From Crowley's (admittedly biased) perspective, demons are not evil so much as doing an unpopular job. He calls Hastur and Ligur nasty enough to pass for human, suggesting that being actively evil the way humans imagine evil is quite uncommon for demons. Crowley himself obviously has no stomach for such things, as evidenced by him needing to go and get drunk for a week after checking out the Inquisition. Notably, his own fulfillment of his duties seems to rely a lot on sounding good (well, evil) and making an impression: he gets commendations for things he didn't do by being in the right area, he speeds because "every little bit helps", and he justifies his approach to his duties by calling it a numbers game: In short, he cultivates an evil image, but the actual evil he does is essentially on the level of widespread, low-concentrated mischief that never does anyone actual harm, like the sigil-shaped highway and reality TV and holding up business phone lines. The few times he does anything concentrated and specific, he takes steps to subvert any evil consequences, e.g. the people he gives guns to who all have miraculous escapes. Let's not forget him reviving a dove Aziraphale accidentally killed while the angel himself is distracted, worrying about the hellhound.
Aziraphale, however, is not privy to this inside view. Presumably Crowley's reluctance to do genuine evil was part of what made the Arrangement seem viable to him in the first place, but he still approaches Crowley with a certain level of caution. Additionally, his description of his side's approach to terrorists and guns ("Current thinking favours them. They lend weight to a moral argument.") - an approach he automatically shares on principle - as well as his willingness to kill the eleven-year-old Antichrist suggest he's more likely to adopt the "the ends justify the means" perspective, even as Crowley is more rooted in the here and now (a contrast shown neatly in the dove scene.) Just as Crowley isn't as conventionally evil as he likes to think, Aziraphale isn't as conventionally good, though they are both to some degree in denial about it and/or trying to keep up appearances.
Thus, the other aspect of his ongoing development in the novel is him getting over the demon thing. He is excited and pleased whenever he finds evidence of Crowley actually being a decent person, as with the guns that magically fail to hurt anyone. His acceptance of Crowley despite him being a demon is helped along by his realisation that good and evil really are only sides, and that Crowley being a demon doesn't really mean anything, any more than him being an angel means anything - in short, that they are Not So Different. When he discovers that Heaven wants the Apocalypse to happen as much as Hell does, he is at once disillusioned of the differences between the two sides as well as clued in that he has more in common with Crowley than he thought, that Crowley is the only one who understands him. When Crowley agrees to stand with him against the Devil, Aziraphale is finally convinced and congratulates him on being a better person than he gave him credit for, while Crowley, in turn, congratulates him on not being as good as he considered himself to be, effectively bringing them down to the same level.
This seals the change in their relationship (and, depending on how you interpret the line about "recent exertions" causing fallout in the form of reality referencing a sappy old love song, may or may not precede yet another relationship upgrade after that, but that's neither here nor there.) It's a process for which Aziraphale arguably needed to change more than Crowley did, the latter having started out as the more grounded (sorry) of the two, likely because of his experience with both sides and his more critical approach in general.
The world is saved despite their bumbling and they could, practically speaking, go about their separate ways again for a decade or two, but that's not what happens. Instead, Crowley offers to "tempt [Aziraphale] to some lunch" and Aziraphale accepts without batting an eye at the obvious reminder of Crowley's nature, in spite of the message of "I'm a demon, I tempt people, I good at it, you could get in trouble for this" barely hidden between the lines - something that would hardly have seemed likely at the beginning of the book.
Aziraphale and Crowley have progressed from chummy but not completely level not-enemies to two friends in equal standing who genuinely trust each other and will be spending more time together - not because they need to or because it's convenient, but because they're happier like that, and acknowledge it, and are fine with that, too.
And now that's out of my system and I really should stop writing and channel that energy into my "Crowley rescues kittens" fanfic or something.
(Thank you for your patience if you've read this far. Feel free to stop fighting the itch and go and ship the hell out of them right now.)
Excellent addendum by Knaccfornerdiness:
You are completely right on every point but I’d like to elaborate a bit on one of them. Both Crowley and Aziraphale choose to see the Arrangement as a relationship purely of convenience. As you said, Aziraphale can’t see it as true friendship because he thinks of Crowley as inherently evil. But Crowley knows they are equals so why does he not see the friendship for what it is? In my opinion, it is because he can’t admit to his attachment to the angel anymore then he can admit to himself that he is kinda a sweetheart.
Crowley lies to himself chronically. Every nice or not so bad thing he ever does is spun just slightly in to a more palatable lie to tell himself. An example of this is when he turns all the paint guns in to real guns and then makes sure no one will really get hurt because “it’s more fun this way.” There are lies and half-truths like that spread throughout his narration, with a hearty collection of them appearing during the parts he spends with Aziraphale or talking about Aziraphale. It is from Crowley that we first hear about how the Arrangement is just a thing of convenience just as it is Crowley who we first see reaching out to Aziraphale. So basically, while Aziraphale’s journey over the length of the text was realizing that Crowley was kinda good and his equal, Crowley’s was one of beginning to accept his own complex nature and all the attachments that entails.
Sorry to add this to your essay but it was so spot on I thought it could use a mention of this.
EDIT: I had a point to add which I just figured out how to say. I actually think Crowley knows to some degree that he is soft but has taken up the internalized self preservation instinct of pretending he's not to deal with having fallen. This theory is based on how he thinks about heaven and hell being the same constantly and the way he words his softer moments