There are actually a couple of interesting questions here. The first is "Can you Chain a Disintegrate?"
You can only Chain a spell that "specifies a single target". Disintegrate doesn't specify any targets at all; rather, it produces an Effect: Ray. Now, you can only aim this at a single object or creature, but that's not the same, in game terms, as "specifying a single target", which would be shown as "Target: One creature or object" in the stat block of the spell. However, Customer Service have most recently been quoted as saying that a ray does count.
I personally wouldn't allow it, especially after the Split Ray text was errata'd to specifically eliminate the "specifies a single target" wording from a feat that is supposed to work with Rays.
Next question - does a ranged attack provoke an AoO?
It's generally assumed that yes, it does... but can you find that written down anywhere?
What you'll find is that the standard action Attack (Ranged) - an action that allows you to make a single ranged attack - provokes an AoO. But the Full Attack action does not. Now, if you use the Full Attack action to, for example, attempt a Disarm, you will provoke an AoO, since Disarm specifically does. But if you use the Full Attack action to make two ranged attacks, you are not using the standard action Attack (Ranged); you are using the Full Attack action. Ranged attacks are not listed anywhere (to the best of my knowledge, at any rate) as provoking AoOs.
The Manyshot feat, while it allows you to make a ranged attack as a standard action, also does not use the Attack (Ranged) action (clearly spelled out in the 3.5 FAQ), and thus, in the absence of text stating that either a/ ranged attacks in general provoke AoOs, or b/ the Manyshot action provokes an AoO, Manyshot does not provoke an AoO either.
Most of your ranged touch attacks take place as part of the Cast a Spell action. This provokes an AoO (unless you cast defensively), but again, unless there is text found stating that any ranged attack provokes an AoO, the touch attack itself does not.
However...
... this means that any archer can use the Full Attack action to fire their full complement of shots - even if that's only one - without provoking an AoO.
You may choose to insert the common understanding into your game, which is the rule that any ranged attack provokes an AoO. In this case, Manyshot, as a ranged attack, provokes an AoO. A Full Attack action firing three arrows provokes three AoOs (from someone with Combat Reflexes, anyway). And casting a spell that requires you to make a ranged touch attack potentially provokes two - one for casting, and one for attacking.
-Hyp.