I can't find a good rule source for whether or not to apply the additional damange dice of weapon special abilities like Holy and Frost for each arrow when using Manyshot.
Conceptually, it makes sense, since the bow should enhance each projectile of the manyshot (and I'd be pretty certain that when using enhanced ammo this would work for sure). I've seen a couple places where people feel this doesn't apply because manyshot says precision damage is not applied (though it is hard to find a good definition of what is and isn't precision damage).
Most sources of bonus damage apply to both arrows
Pathfinder is actually much better about indicating what is and is not precision damage than its predecessor. The rogue's and ninja's sneak attack, as well as the bonus damage from the Shadow Strike and Precision Strike feats (both in the APG), are specifically called out as precision damage. You can reasonably assume that anything not marked as "precision damage" isn't actually precision damage (with respect to Paizo material anyway - it's possible that some 3rd party material should be marked precision damage but isn't) and should be added if it is also not critical damage.
The takeaway is that any other non-critical hit damage bonuses should be added normally for both arrows. Examples of favored enemy and strength bonuses are given, but it also includes the elemental damage from holy, flaming, frost, corrosive, and similar effects as well as any non-precision bonus damage from feats would be added twice.
... but you said "most"...
Some magic properties only trigger on a critical hit, and are thus considered critical damage. Flaming burst and similar properties that added damage on a critical should only be added once, rather than once for each arrow, because of this.