[RPG] Can a Way of the Astral Self monk grapple with its Arms of the Astral Self

No, your Dexterity bonus can't be used for grapple checks instead of Strength.

You don't get the swapping benefit for everything you do unarmed — being unarmed is just the first condition of the ability. You get the effect, when unarmed, for "unarmed strikes and monk weapons."

Starting a grapple isn't striking (aka hitting to deal damage), it's grabbing them to hold on.

The basic principle is that not all melee attacks in the game involve actually hitting someone in the "striking" sense. There are lots of melee attack rolls called for in the rules that are there to resolve something other than a strike.

Now, if you want to throw someone in a martial-arts style after a successful grapple, or grab someone to throw them instead of to impose the Grappled Condition, that's an entirely different ball of wax because grappling doesn't offer that option, but there is another rule that does.

For that you actually want to look at the improvisation rules (PHB, p. 193), which invoke the DM's Contest rules (DMG, p. 238), in order to resolve a throw. If your throw technique involves speed and agility to maximise leverage to use the opponent's own mass and strength against them instead of using your own Strength, that would reasonably qualify as a Dexterity Check for your half of the Contest.

No, they can't be enchanted, because they aren't masterwork and can never be, and all magic weapons must be crafted with a masterwork base.

Creating Magic Weapons

To create a magic weapon, a character needs a heat source and some iron, wood, or leatherworking tools. She also needs a supply of materials, the most obvious being the weapon or the pieces of the weapon to be assembled. Only a masterwork weapon can become a magic weapon, and the masterwork cost is added to the total cost to determine final market value.

It is quite plain that they are not masterwork by default, given the lack of +1 to attack rolls. So let's suppose we want to upgrade them. But it is impossible for an existing unarmed strike to be made masterwork, per this clause in Weapons:

You can’t add the masterwork quality to a weapon after it is created; it must be crafted as a masterwork weapon (see the Craft skill).

All right, let's try making a new one that's masterwork. This fails because unarmed strikes are not manufactured weapons, and you can't manufacture a weapon with the Craft skill if it's specifically not manufactured at all.

Glossary — manufactured weapons:

This category also includes […] in essence, any weapon that is not intrinsic to the creature.

Monk's class features do not help with this, since Craft is not an effect and neither enhances nor improves anything:

A monk’s unarmed strike is treated both as a manufactured weapon and a natural weapon for the purpose of spells and effects that enhance or improve either manufactured weapons or natural weapons.

How about buying one? You might actually be able to buy an ordinary unarmed strike, since they're listed in the Equipment section in the Weapons table, and you needn't pay anything for it*. Unfortunately, per the quote on MW weapons, a weapon must be crafted as masterwork to be MW, and per the preceding this is not possible even for NPCs. There might be an exception, as there seems to be for magic oils, which you can buy in shops but not craft, except that nowhere is "MW unarmed strike" listed for sale.

Magic fang and company do not make a weapon into a "magic weapon"; rather, they give a weapon a +1 enhancement bonus (and the ability to bypass DR /magic because of that), which is not technically the same thing. And while all magic weapons are masterwork by definition as well as by prerequisite, magic fang and friends do not have to be interpreted as giving a weapon retroactive masterwork status, for essentially the same reason, which is fortunate as otherwise the rules would create some bizarre and abusive situations†.

Magic Fang

Magic fang gives one natural weapon of the subject a +1 enhancement bonus on attack and damage rolls.

Magic Weapon

Magic weapon gives a weapon a +1 enhancement bonus on attack and damage rolls. (An enhancement bonus does not stack with a masterwork weapon’s +1 bonus on attack rolls.) You can’t cast this spell on a natural weapon, such as an unarmed strike (instead, see magic fang). A monk’s unarmed strike is considered a weapon, and thus it can be enhanced by this spell.

Magic Weapons

All magic weapons are also masterwork weapons, but their masterwork bonus on attack rolls does not stack with their enhancement bonus on attack rolls.

Special Abilities

Some monsters are vulnerable to magic weapons. Any weapon with at least a +1 magical enhancement bonus on attack and damage rolls overcomes the damage reduction of these monsters.

Note the phrasing of the last quote; they are described as vulnerable to magic weapons, but that's not the rules text, just the plain description, since vulnerability is not defined (except in the incompatible context of energy vulnerability; it should be clear that DR 5/magic does not mean that a +1 longsword does 50% extra damage as well as overcoming DR). The actual rules say that any weapon with a +1 magical enhancement bonus (whether or not the weapon is, per se, a "magic weapon" itself otherwise) overcomes that DR.

(emphasis added to quotes)

* This is, to my mind, patently absurd, but hey, RAW. Monks won't be proficient with those unarmed strikes, but hey, RAW.
† For example, casting magic weapon to increase the price of a club from 0gp to 300gp (permanently? who knows!) and selling it. Or casting magic weapon on any plain weapon to make it masterwork for future crafting. There is a Pathfinder spell for this purpose. It does nothing else‡, is a level higher, costs as much as crafting a MW weapon in the first place, and takes an hour to cast. Allowing magic weapon to do this for free in a standard action is ridiculous.
‡ It has this excellent provision: "If the target object has no masterwork equivalent, the spell has no effect." Doesn't work on unarmed strikes, as they appear to have no MW equivalent and are not objects.