Thematically speaking, it actually makes a lot of sense to mesh the worlds Fairy Tail and Devil May Cry together. Both have themes relating to family (by choice for the former and by blood for the latter), and emphasize the fact that powers in the setting (magical and demonic respectively) are tied to The Power of Friendship and The Power of Love, which are tangible forces in both.
An In-Universe example is discussed in the Edolas arc, when the group are wondering why there are no counterparts for Dante or Modeus. The two explain that Modeus and Trish are from the Demon World, not Earthland, and thus have no Edolas counterpart. Meanwhile, in Dante's case, his father comes from the Demon World and, in his own words, no father, no son.
At first it seems weird that Dante and Modeus would lose their demonic abilities (as well as Dante's Devil Arms becoming ordinary weapons) in Edolas since they're distinct from magic. Except later on it's revealed that magic and demonic energy share similar origins, meaning they likely require similar environments to use.
As an add-on, the reason Nelo Angelo retained his powers is because he was powered directly by Mundus who is stated by Hades to be the source of magic and demonic energy. While this is later proven false, his status as a former god and current ruler of the Underworld means he probably knows better than anyone about how both powers behave, and thus could develop a workaround.
Happy complaining about Lady's weight while carrying her during the Edolas arc is more than just Rule of Funny. Remember, unlike the rest of her allies, she has no magical powers or Hyper Space Arsenal, and has to physically carry her entire arsenal of weapons, which include numerous pistols, grenades, her trademark rocket launcher and extra ammo for each of them. Little wonder the Exceed struggled to carry her, given all the extra weight her weapons would add to her person.
Dante being skeptical of Lucy being his cousin after the Edolas is more than just being afraid to hope for such a lucky break. Everyone's Edolas counterparts are clearly not one-to-one correspondences to their Earthland equivalents, which may include blood relatives. This is supported by the revelation that Faust is Makarov's counterpart and Mystogan's father, despite the fact that Makarov and Jellal are not related at all. Thus, it's likely that not every counterpart will have the same family tree.
Hades maintained that Mundus taught his Eight Kin the lost magics they learned, which include Job's Devil Slayer and Zancrow's God Slayer magics. Given that Slayer magics are based on channeling the powers of the entities they are named after and Sparda's reveal that before he became ruler of the Underworld, Mundus was a god called Pluto, it makes sense that he'd be knowledgeable about how both sets of powers would work.
Lady asking Mirajane for help with her Devil Slayer magic makes a lot of sense when you remember it's explicitly stated that Slayer magic is based on the powers of what they're slaying, and Mira's known for her Satan Soul Takeover i.e. demons. She has experience channeling demonic energy while managing the sideeffects, something Lady would want to learn from.
As for why she doesn't ask Dante or Modeus for help, the two naturally possess demonic energy and likely wouldn't know how to teach someone how to handle it any more than a regular person could teach how to breathe.
Modeus gives an In-Universe example about how Natsu can achieve Lightning-Flame mode. He explains that passing magic on was something that dragons could do, and formed the basis for the creation of Dragon-Slayer Magic (provided it is given willingly).
Cobra learning sound magic is a clever way to cover his bases. Not only does it give him another avenue of attack (at greater range to boot), but the ability to weaponize any sounds his opponent makes prevents any loud noise from damaging his extremely fine hearing. This means he has converted one of his greatest weaknesses into a massive advantage!
After absorbing Gloria's power upon her defeat, Trish guesses it was because as a clone, Gloria's lightning was naturally compatible with her own. A few chapters later, Vergil reveals to Sting and Rogue that Devil Arms can be upgraded by killing and absorbing the power of a demon of the same element as the weapon. Thus, a similar principle probably applies to demons themselves as well.
Vergil stopping anyone from interfering with Dante and Jiemma's fight has an element of pragmatism on top of his usual moral code. Given Jiemma's zero tolerance for failure of any kind (Rogue's thoughts mention he expelled all the members who lost to Dante), he wasn't just forcing his master to follow his own twisted philosophy, he was making sure he wouldn't have a reason to expel even more of his guildmates right after doing so to a promising one like Yukino. While he may not care much for them, he is pragmatic enough to make sure his guilds manpower doesn't diminish because of his "leader's" stupidity.
Bit of fridge humour, but Sapphire becoming a fan of Natsu when he defeated Bora all the way back in Hargeon, i.e., the first chapter/episode of the manga/anime, and having followed his adventures since means she is essentially the equivalent of a real life fan of the series.
