Way of the Dog
(Monk Sublcass)
Certain monastaries in the forgotten parts of the world train in an obscure form of martial arts. Each monk is gifted a puppy of a highly specialized dog breed and the two are raised together. The monk takes care of the pup and trains it himself, forging a lifelong friendship. As they grow up together, the dog is infused with the same ki that its master learns to harness, binding their souls together as they learn to fight in total synchronicity, using one another as an extension of their respective bodies. The result of years of ardurous training is a formidable team and an unshakable bond. Foes are treated to a flurry of combined strikes, bites, and grapples as monk and dog work in perfect harmony to take down any enemy. The core of this Monastic Tradition is the bond between man and beast; with each journeying to greater heights of power and skill, all while learning from one another. As one attains greater strength, so does the other in turn, encouraging one another's path to trancendence…a goal they shall reach together.
Inudo:
Starting at 3rd level, your dog companion is not only your best friend, but an integral component of your martial arts. Your dog is always either the same size as you or one size smaller (up to the player but the dog needs to be able to perch on your character's shoulders or fit between your legs) Utilizing the precise training you and your dog have undergone, your dog occupies the same space as you in combat, positioned either between your legs, on your back, or balanced on your shoulders. Once per turn, you can replace one of your unarmed attacks with one of your dog's attacks. Your dog's attack damage utilizes your martial arts die scaling but the die is rolled twice. Your dog's attacks have the Reach trait and inflicts bleed for 2d4 damage for 1 round, representing advanced techniques where your dog might leap off your shoulders or dart out from between your legs to harry a creature.
Fused Souls:
Starting at 3rd level, your years of training together in the martial applications of ki have bound you and your dog emotionally but it was the tradition of Soul Merging, your monastery's esoteric tradition, that forged yourselves into a single soul, binding your destinies forever. Your dog shares your Hit Points and Armor Class, any attacks or damage directed at it are reflected back to you. Additionally, any later Class Features that directly affect your physiology (i.e. Timeless Body or Empty Body etc.) apply to your dog as well, representing your shared seat at the table of true power and enlightenment.
"Who's A Good Boy?"
Let's face it, your dog is absolutely adorable and everyone knows it. Charisma skill checks (Persuasion) you make have advantage when your dog is within 5ft of you.
Flurry of Fangs:
Starting at 5th level, when you spend a bonus action and 1 ki points to perform Flurry of Blows, you can replace both unarmed attacks with two of your dog's attacks, inflicting bleed for 1d6 damage per turn for 1d4 turns on the target (unless the target is immune). As referenced in Inudo, these attacks have the Reach trait.
Ki-Empowered Bites:
Starting at 6th level, further training and bonding with your dog imparts increased levels of ki into the beast, channeling greater power and accuracy into its bite. Your dog's attacks count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage as well as gaining an additional +4 to hit (you do not gain this attack bonus if you have disadvantage on the attack roll).
Ravage:
Starting at 7th level, instead of using your dog's attack, you may choose to have your dog grapple a creature instead. Regardless of your dog's size, it is always able to grapple Medium and Large sized creatures (A Gnome Monk's tiny dog is able to achieve this through clever manipulation of ki that magically enlarges the dog's jaws to fit its targeted creature.) This works as a traditional grapple (Athletics vs Athletics). While the target is grappled, they receive 3d4 piercing damage per turn, representing the dog's fangs buried in their arm. Any damage dealt to the dog while it is grappling a creature is reflected onto the monk, representing their joined souls. The grapple is broken if you move more than 20ft away from the targeted creature as this prompts your dog to return to your occupied space. If the grapple is broken by the targeted creature, the dog returns to your occupied space. Your dog is vulnerable to attacks of opportunity (and you by proxy) when returning to your occupied space.
At higher levels, the number of damage dice increases by 1 at 11th level 4d4), 15th level (5d4), and 17th level (6d4).
Menacing Posture:
Starting at 13th level, as a bonus action, your dog perches itself atop your shoulders and projects some of your shared ki into a terrifying growl. The target must make a Wisdom saving throw (DC=8+WIS+PROF) or be Frightened for 3 turns. The target creature may attempt to shrug off this status effect at the end of their turn each round by making a Wisdom saving throw. This feature may be used three times before requiring a long rest to replenish it.
At higher levels, the duration of the Frightened status increases by one turn at 15th (4 turns), 17th (5 turns), and 19th level (6 turns).
Astral Pack:
Starting at 17th level, you can spend 5 ki points to have your dog channel its ki to create an astral projection of itself. The astral projection has 5 hit points and its AC is half of your player level + proficiency. The astral projection has its own action and uses the same attack and damage rolls as your dog but the damage type is Force and does not inflict bleed. While the astral projection is present any attack rolls directed at the player or dog have disadvantage and melee weapon attack rolls made towards a creature have advantage. The astral projection lasts for 3 rounds unless it is destroyed, you become incapacitated or die, or you dispel it early. This feature has 3 uses and requires a long rest to regain all expended uses.
Best Answer
There are other answers addressing phrasing and balance, but I want to also review this from the character progression angle as well.
Inudo
At 3rd Level, Monks are expecting to receive a cool new set of actions to take that make significant impact to how they act during combat. Way of the Open Hand gives monks new ways to influence the enemy on hit with Flurry of Blows, Way of the Sun Soul gives monks a ranged attack option, Way of the Drunken Master gives the ability to weave in and out of combat without worry from provoking opportunity attacks.
With your Inudo feature, there isn't any strategic or mechanical changes to the way the Monk will take their turns in combat. They will continue to take the attack action as normal, from the normal range, with no additional options or effects besides some bonus damage. This is boring. I would like to see some new effect or action that involves the dog. It doesn't have to be a combat feature, but Monk does tend to focus on combat with many of its features.
In response to the re-write: With the addition of Who's a Good Boy, the monk now has something unique and interesting to play with even from 3rd level. This gives them new ways to approach encounters, and lets them feel like their subclass matters both in and out of combat. This feels the same as adding a new action because most monks aren't equipped to make Persuasion checks, but now this monk can confidently add it to their toolkit. Great Addition! (again, this review is not examining the balance.)
Flurry of Fangs
Again, this feature does not introduce any new effects or actions for the Monk's turns or overall strategy. I suppose they could use ranged weapons at 10 ft range for the Attack action and then Flurry of Fangs to allow them to step away without provoking Opportunity Attacks. But that's clunky and I'd really be hoping to see some new and unique action that only this special Monk subclass can do. Every monk can do a Flurry, and doesn't feel unique.
In response to the re-write: Having a new type of damage certainly does make this monk subclass feel unique, and having more interaction with the dog does improve the themes of the subclass. Certainly an improvement, but bleed damage should have an explanation somewhere else in your rules. It would make sense to add a few lines to the Combat section of your homebrew rules explaining how Bleed damage interacts with constructs, healing, creatures immune to nonmagical bludgeoning, piercing, slashing, etc.
Ki-Empowered Bites
I can understand why this is here for balance reasons, but it's a little confusing thematically. Is the dog also a monk? Has the dog been training its Ki for 6 levels? Is the monk sending their ki into the dog? I think this needs a bit of flavor text to clear up what this feature signifies.
In response to the re-write: The flavor text you've added to the introduction and this feature clear up any confusion I had originally. You've done a great job linking the themes of the Monk class to this idea of a dog companion. Fused Souls does this even better, by mechanically linking the two. Now it really feels like this dog fits in as a Monk's companion.
Ravage
Finally, we get a feature where the monk can feel unique. They can finally use their dog to execute a strategy that is unique to this subclass, and they have an option on their turn that is different from their usual actions. Now they need to think about how grapples will fit into their game-plan and which enemies to target with this feature. This kind of decision-making is fun for the player and gives a sense of progression as you add more and more options into your toolkit.
In response to the re-write: Still a great feature thematically, and now with a better explanation. I haven't done the math to make any comments about balance, but the numbers look appealing and it makes me want to create a character and try this out in a game.
Menacing Posture
There is no level attached to the description but its position on the list tells me its meant to be a 17th level feature. I like that its a new action that can be taken in or out of combat, and monks don't often have the ability to inflict the frightened condition, so this makes the subclass unique.
In response to the re-write: Now at 13th level, this feels better. It was a strong feature, but maybe a bit lackluster as a subclass capstone. The addition of Astral Pack makes a great capstone for this class, mirroring the supernatural heights that the Monk's main-class features are ascending towards.
Astral Pack
This feature introduces an entirely new strategy in combat, but I can't help but notice that with 5 HP at 17th level... the Astral projection is extremely vulnerable to even a casual blow by a CR10+ creature. The optimal strategy might be to send the dog away at full speed for 3 turns, or have it hide somewhere while the monk makes use of advantage on every attack and disadvantage to being attacked. This strategy betrays the themes of the Monk and dog fighting together as companions. Maybe this feature can be reworked slightly, to reinforce the idea that the dog should stay by the monk's side.
Overall
I think the subclass has some interesting ideas, but I'm concerned that the theme doesn't carry well into the higher tiers of play. By the time the characters reach 20th Level, they're reaching the status of demi-gods and titans. The Monk's final two main class features are "Empty Body" and "Perfect Self". Just the name of those two features give you an idea of how the monk is reaching transcendence and escaping their mortality.
I worry that the themes of a monk and their dog companion don't have the thematic room to grow to such heights. An idea could be to have the dog increase in size at certain level thresholds, or maybe the monk swaps them for a stronger creature at certain levels, to better suit their elevated abilities as they ascend from mortal to immortal. Or maybe you could add some flavor text into the descriptions, giving a feeling that the dog is also on its own journey of ascension.
In response to the re-write: My original concerns have been addressed, and now the dog's place as a Monk's companion is fully explained. The themes presented here show that even at higher tiers, the dog is growing stronger and fits rightfully as the companion to an enlightened monk.