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.

The ring of mystic fire only affects the upcoming spell once

The ring of mystic fire (Magic Item Compendium 125) (7,500 gp; 0 lbs.) does, indeed, grant "a bonus to the amount of damage you deal with the next fire spell you cast before the end of your turn," but the ring only grants that bonus once. Each ray of a scorching ray spell isn't the spell; instead, the whole spell creates multiple rays. The ring improves the spell's damage only once by a maximum of +4d6 no matter how many effects result from the ring-modified spell.

(Ruling that each ray of the scorching ray spell benefits from the ring multiplies the power of the ring by the number of rays—with rather extreme consequences at low levels, especially given that the ring increases also the wearer's caster level for fire spells!)

Although the DM has several options available to adjudicate the ring's effect—allowing the player to distribute the total number of extra dice damage the ring grants among the rays before (or even after) the rays're launched, for instance—, this DM would keep things simple and have the player add the extra damage to his PC's first ray in a volley from the spell scorching ray, in much the same way that precision damage applies to the first attack in a similar volley (Rules Compendium 42). (In fact, I have a fire wizard in a current campaign who's eyeing this very item, and that's how I'm ruling it works.)

Hence, for example, under such a ruling, a level 6 caster wearing a ring of mystic fire can take a swift action to spend 3 of the ring's charges. Then the caster can cast scorching ray, creating 2 rays (because of the increase in caster level granted by the ring) and making 2 ranged touch attacks. If the first ray hits, it deals 8d6 points of fire damage. If the second ray hits, it deals 4d6 points of fire damage.