Hades
Greek God of the Underworld:
Rise from your tomb, mutt, so that I may put you to sleep permanently.
Egyptians were generally buried in tombs, and as a God of Death, Anubis is also associated with Egyptian tombs. Mutt, is obviously in reference to the fact that Anubis is commonly depicted as having the head of the dog. "Putting to sleep permanently" is in reference to animals being euthanized and "being put to sleep".
You were upstaged by Frankenstein, yet you think you can step to me?
During the Middle Kingdom, Osiris actually took over Anubis' job as God of the Underworld, relegating Anubis as Judge of the Undead. The main thing about Osiris' myth that most people know is that his body was chopped into pieces and spread across Egypt. IE, similar to how Frankenstein was put together with a bunch of severed body parts.
I can predict your coffin texts; there won't be any scale tipping.
Coffin texts are collection of Egyptian funerary spells written on coffins. In this case, Hades is calling Anubis' verses coffins texts and calling them predictable. The phrase tipping the scales means to upset the balance in a situation. Anubis was also associated with scales due to his Weighing of the Heart, which involved judging whether or not a soul was fit to go to the afterlife. Hades is saying that there's now way Anubis can turn the tides in his favor.
And I can tell I'm bringing the heat from the wounds you're licking!
This is basically, a reference to dogs in heat. When dogs are in heat they lick their private areas. Also in regards to this, Hades is saying that he's damaging Anubis' genital regions or his masculinity. Bring the heat also means Hades is putting power behind his words.
I'm spitting hellfire, while you're fighting with Styx and stones.
Hellfire is something commonly associated with the Underworld, and a different way of saying spitting fire. Fighting with Styx and stones is a joke on the River Styx, and saying that Anubis is weak.
You don't need to enter my underworld to know that you're boned!
Skeletons are commonly associated with the Underworld, and of course dying, you'll eventually decompose into a skeleton. Boned is also another way of saying screwed, or fucked.
Trust me, this ain't a labor, I'm bringing more harm than is allowed.
One of Heracles labors was to capture Cerberus, a guard dog for the underworld. Heracles ended up asking Hades if he could simply take Cerberus, and Hades agreed, as long as he didn't harm Cerberus. Labor on its own, is also just physical work. Hades is saying this is not a labor, because he's harming this dog that is his opponent, and that it's not a labor because he is exerting any effort.
I'm not a bad fellow, but Zeus knows I'll bring a bitch down!
Hades is often depicted as an evil antagonist to the heroes, when the Hades in mythology was probably the passive of the gods. Zeus knows is simply a replacement for God knows, since Zeus is generally regarded as "King of the Gods". And the main evil thing Hades did was kidnap Persephone, who he made his wife. Granted, Zeus gave permission, but still. And by saying he's bringing a bitch down, he means both in reference to bringing Persephone down to the Underworld, and taking down Anubis.
Anubis
Egyptian Judge of the Dead:
Hm. If you think you're hot, you must be in denile.
The Nile is in actually a very hot river due to being in Egypt. And of course, there's also a joke on "denial". So, Anubis is saying that Hades isn't actually hot, he's just "in the nile/in denial".
'Cause I'm about to fuck up this reject; doggy style!
I don't think this requires an explanation ;P
Step off, your skills on the mic are Tartarusty!
Pun on Tartarus. Anubis is saying Hades is rusty.
I'll leave you in so many bandages, you'll look like my mummies!
Anubis is saying that he'll damage Hades so much that he'll need many, many bandages, and he'll look like a mummy, which is related to Egyptian burial.
You're irrelevant, roman all the place with your claims.
Most of this is related to the next line, but, Roman is a joke on roaming.
Your power's as tiny as Pluto, so it's good you're living up to your name!
The Roman version of Hades is named Pluto. Pluto was a planet, which ended up not being a planet. And then became a planet again, maybe? In any case, Pluto is the tiniest planet, which Anubis is comparing to Hades' power. The irrelevant part of the first line is a reference to the fact of Pluto's planetary status.
Your flow's like Athena, giving me a headache that would cause Dionysus to suffer!
Athena was not born normally, rather she burst from Zeus' head. Obviously, this would cause someone a major headache. Dionysus is a god of Pleasure and Alcohol, essentially, and could be so drunk off his ass that he could ignore pain, except for the headache that Athena bursting from someone's skull would cause. Anubis is equating Hades flow to this pain.
Fall like the Trojans; you have as much chance to win as hell freezing over!
This mostly just set ups the next entrance, but, the Trojans were the main antagonists in the Iliad. Anubis is saying that Hades will fall just like the Trojans did.
Hel
Norse Goddess of Death:
It's Hel, bringing hell, in this dawn of the dead!
Bringing hell is a joke on Hel's name. Dawn of the dead is a reference to the name of the zombie movie, and the fact that this battle has turned into a royale of gods of death.
Bringing Ragnarok on the mic; Lassie and Johnny got no cred!
Ragnarok is the end of the world for the Norse pantheon. Hel is saying that she's going to cause the end of this battle. Lassie is a reference to, well, Lassie. And Johnny is a reference to Johnny Blaze; Ghost Rider.
What's wrong, Hades, you rap like you have Bifrost bite!
A joke on frostbite. Also, people generally shiver and stammer when they're cold, which Hel is saying Hades did in his verse. Bifrost is a rainbow bridge that connects Asgard and Midgard.
Could it be because of Persephone blue-balling you every night?
As stated above, Persephone is Hades' wife. Blue-balling is, well, urban dictionary it if you don't know. Body parts also tend to turn to blue when they're cold, as depicted in media.
And Anubis, I'll be the judge of your lack of rhymes!
As said above, Anubis has basically been relegated to a Judge of the Dead. Thus, Hel is saying she'll turn the table and be judging Anubis' lack of a decent rhyme.
You're not sick, but I'll make an exception for you in my domain this time!
Hel's underworld was relegated to those have died of old age or sickness. And "sick" in this case is referencing the rap term; sick as in being at rapping. Hel is saying that despite Anubis not being sick, she'll still bring him down to her underworld.
My words may be cold like Ymir, but I'm still burning you hotter than Surtr!
Ymir was the father of the frost giants, and Surtr is a being of pure flame. Hel is saying despite her words being ice cold, she's still burning everyone hotter than a being of fire.
Your boasts are baldrdash; when you fall, no one will shed a tear!
A joke replacing "balder" in balderdash with the name of the god Baldr. This also sets up the next bit. Basically, Baldr died, but had the potential to be brought back if everyone cried. The only person who didn't was an old woman, who was Loki in disguise. Hel is saying that when they die, no one will cry to try and get them back.
Izanami:
Japanese Queen of the Dead
The queen of the dead is here to reign in on all your celebrations!
Izanami in Shinto mythology is known as the Queen of the dead. Reigning in this case is a double meaning of reign/rain. She's both ruling over everyone, and raining on everyone's parade.
I can predict all your actions; you'll send yourself to Yomi in frustration!
This is again a dual meaning. Yomi is a game by Sirlin, meaning "Reading the mind of your opponent", which ties into predicting everyone's actions. Yomi is also the Japanese Underworld. Izanami is saying that her opponent's will end up killing themselves out of the fact that Izanami is unbeatable.
Listening to all of you rap is worse than labor, but I'm heating things up even more!
Izanami ended up dying during giving birth to a fire god. Izanami is basically saying everyone's verses are worse than childbirth, but she's still heating things up more than when she gave birth to said fire god.
I'm killing all these verses, and none of you will be able to create more!
When Izanagi, her husband, sealed her away in Yomi, she claimed that she would kill 1000 people each day. To which Izanagi replied that he would create 1500 people each day. Izanami is saying that she's killing these verses (slang for doing good), and that her opponent's won't be able to create any more verses once she's done.
You're all just products of my creation, so roll over Anubis, and everyone leave!
Izanami is also a creator god in Shinto mythology, and she's saying she created everyone here. Roll over Anubis is a reference to Anubis looking like a dog, as well as a reference to when Izanagi rolled a boulder over the entrance to Yomi, sealing her in and left her. Izanami is saying Anubis is as dense as a boulder, and that everyone else should leave in fear of her.
All of you are fishing for ammo, but we're the ones who are still being believed!
Izanami is saying that everyone else is fishing for something to say against her, but are failing. Also a reference to the fact that once of the first gods Izanami gave birth to was a fishing god. The second part is Izanami saying that she's the most important and relevant in this battle, because Shinto is still being worshiped as a religion today.
Baron Samedi:
Voodoo Lord of the Dead
It's the baron, bringing class and life to this funeral!
Baron is a class of nobility, and Samedi is known for being quite jolly despite what he's associated with. Thus, bringing life. Also a joke about the fact that he's bringing "life" to a funeral. The funeral in this case being all of his opponent's.
All you boneheads, listen up, as I make your bones shake and rattle!
The bone puns should be obvious. Baron Samedi is commonly depicted as a dapper skeleton. Shake and rattle is a term for being afraid, but it's also a reference to the blues song "Shake, Rattle and Roll".
Anubis and Hades, why so serious? I prefered the two of you on Disney.
Samedi is calling Anubis and Hades out for their "grimdarkness" approach to death, compared to Samedi always try to be the life of a party. And the second bit is a reference to the fact that both Anubis and Hades starred in Disney products. Hades in Hercules, and Anubis in Kim Possible.
And yikes, Izanami and Hel, who'd Voodoo either of you to make babies?
Samedi is basically calling Izanami and Hel horrifying, question who'd ever have sex with them to make children. (The Voodoo is a joke on the fact that Baron Samedi is a Voodoo god, and is using "Voodoo" as a replacement for "do".
I've got friends on the other side, and even they're lighter than you.
Friends on the other side is a reference to a song sung by the villain in Disney's Princess of the Frog, Dr. Facilier. Facilier uses Voodoo powers, and was in fact inspired partly by Baron Samedi. The friends in this case are dead people, who Samedi is saying are capable of having more fun than the four of his opponents.
Yeesh, at least dress sharp, I'll have to Loa-n all of you a funeral suit.
This is a reference to the fact that Samedi is commonly depicted as wearing a very fine suit and top hat. Loa are Voodoo spirits, and this is a joke on the word loan. Funeral suit, in this case, refers to the suit that dead people wear during a wake. IE, Samedi wants everyone to look nice as they die.
I got more flow than Agwe, and trust me, you're all meeting only one face of Erzulie.
Agwe is a Voodoo god of the water, and Erzulie is a Goddess of Love and Beauty. The three faces of Erzulie are Lust, Compassion, and Loneliness. Since Lust and Compassion obviously wouldn't fit as a insult, Samedi is basically saying they'll always be alone.
Let me just snuff out all the candles, 'cause this about to get unruly...
Candles are an object commonly used in Voodoo practices. By snuffing them out, Samedi is implying that what he's about to do to these is too cruel for light to be shown during it. Alternatively, he could be referring to candles as everyone else's soul. Claiming he's going to kill everyone else involved in this battle.