Yes, magic from the arrow and the bow combine, and stack as long as they are different.
You can see this in the description of the various special properties themselves:
Flaming [...] Bows, crossbows, and slings so crafted bestow the fire energy upon their ammunition.
Flaming Burst [...] Bows, crossbows, and slings so crafted bestow the fire energy upon their ammunition.
Frost [...] Bows, crossbows, and slings so crafted bestow the cold energy upon their ammunition.
And so on.
Not every special property says this: dancing doesn’t (which makes sense since it is melee-only), defending doesn’t (since the only thing it does is change the weapon’s enhancement bonus, which already gets transferred to the ammunition by default), but pretty much every special property where it makes sense, has this line.
It’s so common, in fact, that I would treat any case that neglected to mention it but where it would make sense as an error, and assume that could be transferred as well. I would also ignore the fact that the rules call out bows, crossbows, and slings; in the case of some other kind of projectile weapon, I would allow that too. Really, it should just have been a general rule, but for some reason they didn’t do it that way. Perhaps they were worried about a defending bow draining a +5 arrow of its enhancement and gaining AC? Should have just called out defending as an exception there, in my opinion, but oh well.
Same source bonuses still don’t stack, however. If you shoot a flaming arrow from a flaming bow, the +1d6 fire damage from the one instance of flaming (say, the bow’s) does not stack with the other instance of flaming (the arrow’s, in this case). Likewise, bonuses of the same type don’t stack: this is spelled out explicitly for enhancement bonuses, but it’s still true for other bonuses. So if two different properties gave, say, sacred bonuses to damage, found on the bow and on the arrow, the final bonus would be the larger of the two.
