Find one as treasure / Go on a quest for it

By the book is a little odd to ask for here — the place to find an adamantine weapon is in the game world, not in some rule in the PHB that would give players unfettered access to powerful treasure.

For a veneer of by-the-book analysis, unenchanted adamantine items are listed as magic items in the DMG (pp. 146–8, 150). There's only armour listed, but adding an adamantine weapon to a treasure trove would be a trivial bit of homebrewing (that many DMs wouldn't even consider homebrewing).

So the answer is: go on a quest for adamantine, either in weapon form, or raw material form and then a second quest to find the legendary smith who can work “one of the hardest substances in existence.” Or mention the desire to your DM, and get lucky finding it in a treasure cache. That's how you get an adamantine weapon.

1. 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.

2. 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.