Just by looking into him carefully, you can learn a lot about Azrael:
Azrael's weakness to darkness themed attacks may seem a little random, or a bit generic given that he's an angel. But then you learn he's Phasmaphobic (scared of ghosts).
Azrael's snarkiness may also seem random. Then you remember that he had to live with Azazel for about 13-14 years before they died. And even before he reunited with him, he still had to deal with some of the wackier characters in Inferno, like Abigail and Wilhelmoku. And after that, he was taken in by Thor, who can be summed up as 'Azazel but a Nice Guy'. No wonder he's such a Deadpan Snarker! Poor guy's been the Only Sane Man his whole life!
His Nightmare Fetishist nature can be explained as well. Given that Azazel likes talking about blood and killing, and he had to deal with both this and the deaths in Inferno, one might say he's been Conditioned to Accept Horror.
Though minor, Azrael uses Shock and Awe attacks. His battle style is also the most technology-focused, and his first Leitmotif, Hexagon Force, is inspired by 80's themes and Sega Genesis music. Then you remember that he's the gamer of the group, if President Evil is any indication.
Most Leitmotifs are actually fitting for the characters they're on. Let's go over them:
Azrael's Hexagon Force, as mentioned above, is an electronic theme resonating with Azrael's electric powers. However, it is also somewhat heroic and inspiring, fitting Azrael's status as a Nice Guy and All-Loving Hero.
His arguable second theme, To The Skies, represents his second element of Light. It is also an angelic sounding theme and quite soothing, and Azrael is a very relaxed angel.
Freya's Swirl is a very upbeat and cheerful song. She is not only a Love Goddess, but is also one of the less mean characters in the book (even though that doesn't say much).
Azazel's Clutterfunk is very wild and loud, just like him. It's also one of the more 'alien' sounding themes, which fits his status as the only demon in a group of angels and gods.
Uriel's Theory Of Everything 2 is a very quiet theme that sounds completely fitting for the stoic genius that Uriel is. Why To E 2 was used instead of the original is because the original Theory of Everything was used for Inigo in The Spectrum Game, who fits the 'stoic genius' description better.
Forseti's Time Machine requires some thought, but he's the resident ice mage, and the theme gives off an 'icy' vibe. It also sounds rather logical, given that he is The Spock among the gods.
Baldur's theme is called Adventure Fantasy. In Chapter 7, he had a fantasy about adventuring alongside Azazel in their campaign against Helheim.
Why is Azazel so intent on killing things? Well, being sent to Helheim and forced to work for Hel may have desensitised him to such morals. Why he's still starkly a good guy can be summed up to Heroic Willpower: Azazel isn't the type to just give in to evil like that. He may have loosened his morals slightly, but overall, he's still a hero.
Why does Iceaura try to shoot Azrael over some bad puns, but not do so when Uriel does the same? Because he's had more time with Azrael's puns. He's clearly sick and tired of it.
The reveal that Azazel evolved his Aesir Cross before the story began shows that he clearly has a lot of self-awareness, which is why his counter-argument to Uriel's calling him out on his hypocrisy in Chapter 9 chalks up to At Least I Admit It.
