does the character add the strength bonus of the bow to each arrow?
Yes. Manyshot prevents precision damage from applying multiple times, but doesn’t do anything about other damage bonuses. The Strength bonus on damage rolls is not precision damage. (Precision damage is stuff like sneak attack, sudden strike, skirmish, or the duelist’s precise strike.)
If the bow had magical properties like cold damage, would this apply to each arrow?
Yes, unless that damage is precision damage. The assassination property from this Cityscape web enhancement and the deadly precision property from Magic Item Compendium are the only weapon properties I know of that add or increase precision damage.¹
Does the same apply if the character uses split arrows? (enchant)
Your question title refers to the conjuration arrowsplit, while this mentions “enchant,” which I am guessing refers to the splitting weapon property found in Champions of Ruin.
The answer to both is the same: yes, these copy all features of the split arrow, including the Strength bonus to damage as well as any special weapon properties found on the arrows themselves or imparted upon them by the bow.
- Thanks to @Prevarications for finding these two.