Between the boot-up cutscene and the title screen, the moon forms the number "0" just before Dante's silhouette slashes it to form a "3", displaying the game's full title. The zero serves as a reminder of this game being a prequel to DMC1. It's a neat but very subtle way to demonstrate the Episode Zero: The Beginning trope. It may also allude to a meta tidbit, as a passage from page 211 of the 3142 Graphic Arts artbook interviews mention "DMC 3 has even been called DMC 0 at times".
Temen-ni-gru, the "foundation that brought out fear", shares its name with the Great Ziggurat of Ur (a.k.a. "Etemenniguru"). It's either Capcom are just picking names out of a hat, or that Dante lives in Iraq. Coincidentally enough, the city, as seen from atop of the Temen-ni-gru, is surrounded by a vast desert.
One may wonder why Cerberus, a hellhound, is an ice elemental. Anybody who has read Dante's Inferno (or at least studied up on it), however, would know that in the ninth circle, Satan was encased in ice. Another nod towards the epic, perhaps?
The third circle of Hell in The Divine Comedy, where Cerberus is the guardian, is mentioned to constantly be rained upon with cold, miserable slush. DMC's Cerberus may be a more direct reference to that. And the mission on which the fight against Cerberus takes place? Mission 3.
All bosses possess a Logical Weakness. Agni and Cerberus are weak against each other, for instance. A notable exception occurs during Dante's campaign, as none of his weapons deal extra damage against Beowulf. Vergil does not have that difficulty, as Yamato's darkness counters Beowulf's light. That explains his quick end by Vergil's hand.
The Gigapedes in the Leviathan's intestines are a Degraded Boss when compared to the previous Gigapede (which is fought as a proper "boss" battle). However, they can still be justified from another perspective. Apart from losing a dedicated boss lifebar, the game just likely wants to make you feel as though they are degraded; the "weaker" Gigapedes are unable to use some of the boss version's varied electric attacks because they are moving in very cramped intestine "hallways". And because their heads are always sticking out when chasing you from behind, it's inevitable that you will always Attack Its Weak Point if you decide to fight it, justifying why they can be taken out faster.
Doppelganger the Deathvoid is described as a shadow demon that takes the same shape as his foes. However, the symbolism underlining the fight can be viewed as a subtle use of the Enemy Within trope (particularly, understanding that the shadow represents a part of the person they must overcome). Dante even lampshades said trope by saying "I know why you're here. You want to ask me some questions. Well, too bad. I've already answered them myself. I don't need you anymore." as soon as he meets the shadow demon. Doppelganger initially taking on the appearance of Dante's Devil Trigger form is an understandable visual representation of the darker side within Dante. On the other hand, after Doppelganger has been defeated and repurposed as Dante's Doppelganger Style, the clone takes on Dante's human appearance instead, which could visually represent the symbolism of it being "tamed".
Throughout the story, Dante consistently refers to his parents as "my father" or "my mother", completely foregoing the pronoun "our" when referring to Vergil or even speaking to him in person (e.g. Dante saying "So my mother's amulet is the key that unlocks the door to the Demon World" to Vergil's face). He only acknowledges their shared parents when explaining things to Lady, and in his final battle with Vergil. This shows just how deep-seated Dante's disdain for Vergil is, so much that he tries to ignore Vergil's connection to their parents.
Unfortunately, part of Vergil's trauma is his belief that their mother never came to find him on the day their home was attacked. This means Dante's wording regarding their parents is alienating Vergil further and fueling his bitterness.
At first glance, Vergil, who is established as being an honorable Anti-Villain, stabbing Dante through his chest would seem as utterly lazy if not terrible writing in seeming to inexplicably turn Vergil into something of a Card-Carrying Villain out of nowhere. But when one considers that Vergil and Dante are well aware of their partly demonic heritage, of which includes the benefit of a Healing Factor, it would be obvious that Vergil would know that he wouldn't be able to kill Dante that easily and pierced him just to keep him down so he could leave with Dante's half of the Perfect Amulet. He also likely wanted to force Dante to awaken his Devil Trigger powers to help him survive. Vergil's "Might controls everything" speech also implies that he wants Dante to become stronger, and the Devil Trigger form is the first step to that.
Vergil stabbing Arkham with Yamato, intending to kill him now that the final door was open, seems harsh. However, it's not just that Vergil was done using Arkham - beforehand, he accused Arkham of feeling sympathy for his daughter, Lady. The moment Vergil had the slightest feeling that his accomplice was letting love touch his plot for power, he had to literally cut him out, lest Vergil allow himself to falter and feel any sort of love for Dante too.
Lady gives her rocket launcher Kalina Ann to Dante hoping that it will help him defeat Arkham. It's subtle, but it does make sense from a gameplay perspective with enough context. In his blob form, Arkham has a habit of summoning creatures (called in-game as Legions) that swarm and circle around Dante. The Kalina Ann is the most effective weapon to use against these Legions due to its high damage and splash radius capable of eliminating several of them at once.
In the penultimate boss battle, Dante and Vergil act in unison as they hack the blob-turned Arkham, and then toss over their sibling's weapon back to each other. It's even more noticeable when they deliver the killing blow as they shoot Arkham and shout "Jackpot!" at the same time. Not only is this scene awesome, perhaps there's a subtle credence to their unison, as twins in real life tend to (or are commonly believed to) have the same thoughts and do the same thing at the same time.
In the bonus post-credits scene, Vergil's act of throwing the Yamato's sheath before charging at Mundus implies that he knows his chances of winning against the Demon King are slim, yet won't back down from his opponent. Those who have played the first game know that Vergil's defeat is a Foregone Conclusion because he will be enslaved as Nelo Angelo. But think about it in a different perspective; Vergil is an Iaijutsu Practitioner who relies on unsheathed attacks and even performs a Sheath Strike for his basic combo, so he actually handicapped himself by doing this.
Lady's (understandable) attitude is pretty nightmarish in itself on a more mental level. Here is a woman who had everything taken from her, including (metaphorically) her name. Unfortunately, her disregard for her own life and others during her Roaring Rampage of Revenge is almost sociopathic - A signature scene is when Dante catches her as she is falling from the tower, with Dante holding one of her legs as the only reason she isn't still plummeting. Her first response is to tell him to let her go, while we find out a few moments later that she can save herself. She has a clear disregard for her own safety, but the worst is when Dante doesn't comply, she shoots him in the head (at that point she did not know Dante was a Human-Demon Hybrid). That's right, if it wasn't for Dante not being quite human enough to survive a shot to the dome, she would have murdered a person only trying to help without a hint of regret.
(a.k.a. "Etemenniguru"). It's either Capcom are just picking names out of a hat, or that Dante lives in Iraq. Coincidentally enough, the city, as seen from atop of the Temen-ni-gru, is surrounded by a vast desert.