An attempt at making the very reddest deck possible. I think I'm pretty close. Needs a few things, like Gauntlet of Might, but I'll never be able to afford that card or Candelabra of Tawnos. This deck does all the Red things you love; bursts of mana, wheeling, direct damage and indirect mass murder, and of course, cheating big hasty critters. All it's really missing is chaos and cruel control, both of which I greatly enjoy, but I recommend Norin the Wary for those approaches. You can't argue with TappedOut tradition.
Some notes on specific aspects of the deck:
Mana Tricks: Mana Flare, Caged Sun, and Gauntlet of Power all increase your (and others's) mana greatly. Runaway Steam-Kin and Birgi, God of Storytelling
provide regular, though smaller, extra mana, and don't forget Irencrag Feat, Jeska's Will, Rousing Refrain, or for that matter Koth's minus ability. Then there's the Nehebs, and best of all, the Leyline Tyrant, which makes all the mana carry over from phase to phase and turn to turn. This makes Braid of Fire good for something other than instants and your instant-speed activateds, which are : Ashen Firebeast, Cavalier of Flame, Granite Gargoyle (though an unexciting mana sink), Purphoros, Bronze-Blooded, Inferno of the Star Mounts and Jaya Ballard, Task Mage.
Cards, cards, cards: There are 5 Wheel effects in this deck (counting Ruin Grinder's death trigger), and a good bit of impulsive draw; Birgi's backside is fantastic for it, Neheb, Dreadhorde Champion provides another repeatable massive rummage (and Cavalier of Flame a one-shot), Ox of Agonas is a mini-wheel. All of those wheels will absolutely trigger any opponent that isn't playing a graveyard deck, and blue players in particular fucking hate them. This, of course, is more reason to use them.
Free casting: Sunbird enables endless free casting, Ilharg, the Raze-Boar and Purphoros are a dynamic duo of cheating, and Electrodominance will give you a free spell. This is more free casting than you need, with the deck's ridiculous mana output.
Bringing the pain: Of course, Toralf is the commander, and this deck is well-equipped to take advantage of his trigger. Blasphemous Act, Starstorm, Immolating Gyre, an overloaded Mizzium Mortars, and Star of Extinction (and likely a couple others I'm forgetting) can all automatically end the game if your opponents have any amount of creatures out when Thor's at the party. His synergy with Torbran, Solphim, and Dictate of the Twin Gods is ridiculous, and a field of 1/1s with Toralf and Torbran and even the slightest bit of direct damage is a hilarious sight to behold. Form of the Dinosaur can yield extreme overkill damage once a turn, but your life will be hurt. But if you were planning on caution, you shouldn't be playing Red.
Additional rapid death: With all of the mass-murder effects you have, there's a very good chance that you'll occasionally be able to swing in against undefended opponents, especially with the good number of haste enablers. Firecat Blitz is great at this too. Terror of Mount Velus can set up the kill well, but you got one chance at it.
Some additional comments: the usual approach with Toralf is to really lean into making his trigger as unpleasant as possible, usually by giving him death touch and often indestructible and/or lifelink, e.g. with Basilisk Collar. This is a fine strategy, and I generally recommend it with Toralf. However, this deck isn't really intended to maximize Toralf per se, as much as it absolutely capitalizes on him and enhances him incidental to doing red things. Some cards I might cut if I were to lean into that aspect of him would likely include Terror of Mount Velus, Mountain Stronghold, Urabrask the Hidden, and maybe Yidaro.
On the note of other cards you can safely cut if you want to make this more competitive, Granite Gargoyle comes readily to mind. It's mostly in here just because I really like the card from when I started playing. I realize my nostalgia is not yours, and I would generally recommend not using him.
If you're proxying and/or money is no object, definitely include Gauntlet of Might and possibly Candelabra of Tawnos, which is great once you have a doubler, but not so great otherwise.
There's a number of ways to make this crueler, and I generally support them. Blood Moon is an obvious inclusion, but I'm low on people that I regularly play against, so to not antagonizze them further, I don't use it (regularly). Obviously consult with your group if you aim to use it. If you do use Blood Moon or Magus of the Moon, I may recommend cutting a couple of the utility lands, but there's not that many as it is.
In general, this is not a hyper-competitive deck, though it is very capable of doing amazing things while being pretty much a thrill-a-minute. I do believe it can be made far more competitive while remaining a celebration of Red without much effort, and my above notes should provide ample guidance on this.
