Your pricing of the small masterwork light repeating crossbow is correct.
Your pricing of the continuous master’s touch is correct per the guidelines. As a DM, I might not be so sure I like that price, however.
Your pricing of the at-will use-activated magic missile is correct. However, it would not fire missiles that deal 3d4+1 damage, as you seem to imply. It would instead fire three separate missiles that deal 1d4+1 damage each.
You have a mistake when you add them together: firstly, you dropped the \$\times2\$ multiplier from the duration note on master’s touch: it should be \$10,000\text{ gp}+4000\text{ gp}\$, not \$+2000\text{ gp}\$. Moreover, you missed this note:
Multiple Different Abilities
Abilities such as an attack roll bonus or saving throw bonus and a spell-like function are not similar, and their values are simply added together to determine the cost. For items that do take up a space on a character’s body each additional power not only has no discount but instead has a 50% increase in price.
For one weapon with two different effects, you need to pay half-again the cost of the continuous master’s touch. So the cost should be \$10,000\text{ gp} + \left(4000\text{ gp} \times 1.5\right) = 16,000\text{ gp}\$.
Finally, Eberron Campaign Setting has a page (268, to be precise) on warforged components. It includes statements like
A warforged component usually occupies the same space on the body that a magic item of the same kind normally would.
Likewise, armbow specifies that it
attaches to the arm of a warforged, completely covering the hand.
So assuming that this item uses up your character’s hand like the armbow does, it should cost \$16,000\text{ gp}\$. If it does not,
Components that do not occupy any space on the body cost twice what they would cost as ordinary magic items.
Thus that would be \$16,000\text{ gp} \times 2 = 32,000\text{ gp}\$.
Either way, the fact that the base cost is less than the price of the armbow, despite having significant extra features, suggests to me that your DM may very well dispute these prices. I would.
A magic weapon must have a +1 bonus on it before receiving any other special weapon properties. But you aren’t enhancing this as a weapon, you are treating it like a wondrous item. The rules don’t really cover this possibility; I would probably require the +1. But a +1 is all you need; after that, you’re free to put as much other magic on there as you want without needing a higher enhancement bonus (with the exception of the more powerful weapon augment crystals).
Any enhancement bonus on the item would add to attack and damage, yes.
For costing these, the fact that you’re adding features to a magic weapon means that you need to consider that 50% premium again. You have a \$10,000\text{ gp}\$ component (the magic missiles), a \$4000\text{ gp}\$ component (the master’s touch), and a \$2000\text{ gp}\$ component (the enhancement bonus, assuming you go with the minimum +1). The second and third each cost 50% extra because you’re combining them on one item. Like so:
$$ 10,000\text{ gp} + \left(4000\text{ gp} \times 1.5\right) + \left(2000\text{ gp} \times 1.5\right) = 19,000\text{ gp} $$
This adds onto the \$550\text{ gp}\$ cost of the base item, so your total is \$19,550\text{ gp}\$. Again, twice that if it’s not taking up your hand.
Best Answer
RAW is problematic on this point. I am therefore going to address this question in a rather unusual order for a rules-as-written question: I am going to discuss the (highly-likely) intent, and then discuss what the rules do or don’t say and why they are problematic.
The intent seems pretty clear: the masterwork benefit is included in mithral’s
In a very-unusual case, I believe we can safely determine the intent of how mithral should be handled. We have examples of specific items statted up in the Player’s Handbook, including the elven chain and mithral shirt. These appear (particularly in the latter case, since it doesn’t have the fancy name) to be by-the-book examples of the mithral special material being applied to the chainmail and chain shirt items, respectively.
In both cases, the armors are statted with the masterwork bonus not stacking with the benefit of mithral.
It is important to note that these examples do not provide rules for applying mithral. They are, at best, secondary sources for what it means to be mithral, and specific items can and do provide exceptions to general rules; it is entirely possible to have specific items differ from the items you would get by applying the modular pieces the item appears to be made of.
And even in particularly blatant cases like the mithral shirt, examples are not the primary source for the general rule. In a contradiction between the two, the errata rules would have us ignore the examples and follow the primary rule.
Thus, we can only make an argument for intent on the basis of these examples, but I argue that in this case, they provide a very strong argument for intent.
The rules fail to describe this, however
Because the examples above give a very strong indication of the likely intent of how mithral should be applied, we expect the mithral rules themselves (the primary source, now), to describe the process such that we get the results indicated by the examples, where the benefit of mithral supersedes or includes that of masterwork without stacking.
In particular, one of these would suffice to get the results we expect:
A statement that says that the benefits described for making an armor from mithral already include the benefit for being masterwork.
A statement that says that these two benefits should not stack with one another.
However, the rules say only that the cost of being masterwork is included in the cost of being mithral; they do not say whether or not the benefit of being masterwork is included in the benefit of being mithral. That’s one option down.
Furthermore, the general stacking rules are no help, even if we accept that the reduction of ACP is a bonus subject to the bonus-stacking rules,1 since the benefit from both masterwork and mithral are untyped, and they are separate sources. Under the usual stacking rules, these two benefits would stack.
Conclusion
Super-strict RAW, two separate benefits are described, they are not given types, and there is nothing in the rules that says they should not stack. However, available examples strongly suggest that this was an oversight, to the point that I would consider it dishonest to fail to mention them.