I'm always glad to see some appreciation for the pseudo-"Eastern" classes that Gary threw in for fun, so here are my notes, in the spirit of being constructive. My perspective is that each class ability should be something you can connect directly to background or use to flesh out a persona. Since this is a custom class, you can really tailor it to the individual character!
I took a try at creating one, tossing most of the magical-like abilities. I did not want to step on the warrior...
Can you say why you decided to remove magical abilities? Doesn't that just create a martial artist instead of a monk?
Alignment: I agree with NobleDrake
that you should let the player choose an alignment, and work with them to flesh out the monk's background. Chaos Monks would be very interesting indeed, if you and your player can work out a way for the PC to be able to walk around a village or town without being arrested on sight.
Weapon training: Given the focus will be on unarmed combat, I'd restrict a 1st level monk to only staff, longbow and temple sword (a finely balanced weapon used in training, and the PC has -2 to use "western" longswords until he quests to find a sword he's comfortable with). Gain proficiency in another "western" type weapon from your proposed list at each level, as the monk learns more about his new country (and trains incessantly while the Warriors and Thieves blow their gold on carousing).
Armor: It's a little ironic that monk characters are serene and peaceful, while their players are a bundle of nerves wondering if the NEXT hit will be the one that gets through their crazy-high AC and demolishes their meagre pile of hit points. But until then, monks will be untouchable. I bet as soon as your other players see this monk in play, the only characters surviving the funnel will be the AGI 16-18 crowd, who will all become monks.
I don't quite know how to fix this, but I would investigate something along the lines of
monks don't know armor, don't use it, and none of their abilities (including the following) work at all when the monk wears any protective gear
starting AC of 11, or 12, or 13
gain bonus to AC = class level when wearing robes (not tunics or western clothes)
gain double the agility modifier when unarmored, and while they can still be surprised, they can always use their agility bonuses
start with 2D6 hit points at level one, gain 1D6+1 every level after that
massive armor check penalties to acrobatics (and combat) when carrying anything more than one staff, one burlap sack with a bowl and blanket.
That last one is where the DM can prevent the monk from becoming a munchkin in the hands of a power gamer. Yes, he might be strong in combat, but the party will quickly get tired of him having to drop his share of the treasure to fight off a giant snake, and the coins trickle down a crack in the cavern floor...
Attack (Deed Die)/Crit Die, etc: I'm fine with them being as powerful as warriors, if not as versatile - a monk can't pick up any mace or flail he finds lying on a rotting pew in the old church and smack a skeleton as hard as a warrior could. But with the right weapon, they should be able to fight better than clerics or thieves.
Mighty Deeds: This is the major issue I have. My opinion is that Deeds should be restricted to Warriors and Dwarves as an iron-clad rule. I think letting other classes use Mighty Deeds is as bad as letting other classes cast wizard spells. Copying a Warrior's best ability is what would "step on him" the most.
I'd replace it with the funky quasi-mystical abilities, and in the same way a rookie Warrior isn't sure what his Mighty Deed attempts can accomplish, I'd keep much of the Monk's potential hidden. The player won't know what his PC is capable of until he discovers it in an adventure. You can tie it in nicely as "the Monk's journey is one of self-discovery and enlightenment". He'll have some great stories: "Remember when I discovered I could swim underwater for 10 minutes with a single breath of air?" "Remember when my monk used pressure points to paralyze that evil wizard's wand-waving arm?"
Unarmed Combat: with a (non-Deed) Attack Die adding 1-3 points of damage, the monk is quite powerful. I'm fine with how you've done that. And if it turns out to be too strong, it's an easy fix - the PCs start encountering non-human monsters with no pressure points and stony skin. All of a sudden, the Attack Die doesn't add to damage anymore.
Luck: A monk's Luck modifier applies to all unarmed attacks.
But not unarmed damage, right? Just the "to-hit"?
Fast Movement: I'd change the monk's movement to 30 + (levelx5) feet when unencumbered. If the player is lucky enough and skilled enough to reach high levels with a Monk, his reward is Super-speed. Give him red robes with a yellow lightning bolt on the front.
Dodge Weapon: I wouldn't let a monk's speed trump the Magic Missile spell. And I would say that a monk can attempt a saving throw to dodge the FIRST missile attack against him each round.
Falling: No issues here, I just want to point out that this is a great spot for a monk's hidden potential to shine through, at whatever class level you deem appropriate. A Reflex save or Agility check to stop falling (suddenly, you grab a branch growing out of the side of the cliff! There's a strawberry growing from a small plant directly above you!)
, another check to grab a comrade falling next to you, etc.
Healing: I'd suggest that any monk who medititates for 2 hours at the end of the day heals 1d12 hit points. It's not a source of healing during the adventure
, but it does create an incentive for the monk's party to find a safe room in the spider cult's temple where they can spike the door and rest in the night, old-school style.
Grappling: This is where I might possibly be persuaded that a monk could accomplish a somewhat-mighty deed. Using a judo move to fling someone across the room into a tangle of rotted curtains, and so on.
A lot of the flavour of a Monk can come from the acrobatic flips and leaps the player describes - I'd let a Monk who can't cast spells, has no Deeds, no thief's Luck, no Feign Death
etc, to get away with a lot of great maneuvers when the player roleplays it well. Doesn't that fit in well with the old-school rule of "you can accomplish anything that you can convince the DM your character would be able to do"?
A monk may not be able to disable or dismember a monster as a Warrior might, but he should be the most mobile fighter in the battle, and have a pretty good chance to avoid being trapped, surrounded, knocked prone or pinned. In abstract DCC combat, all it takes is some imagination on the part of the DM.