Let's apply the rules literally to your examples and see what happens.
Tail spine attack
Suppose a monk uses Deflect Missiles to reduce the damage of a spined devil's tail spine attack to zero.
Tail Spine. Ranged Weapon Attack: +4 to hit, range 20/80 ft., one target. Hit: 4 (1d4 + 2) piercing damage plus 3 (1d6) fire damage.
This is a ranged weapon attack, so it can be deflected. The devil is small, so its spine is certainly small enough to hold in one hand, which means the monk catches it. They then spend one ki point to throw the spine at an enemy. As described in the monk ability, for this attack, the spine is considered a monk weapon, conferring all the benefits that go along with that: the monk makes uses their dexterity modifier for the attack and damage modifiers, and they get to use their martial arts die in place of the "normal damage" of the weapon. Other than that, the missile has all of its normal effects, since nothing about the monk's Deflect Missiles ability says otherwise.
So, the main ambiguity here is: what is the "normal damage" of the spine? I believe we can conclude that it is the d4. The damage is listed as "1d4 + 2 piercing damage plus 1d6 fire damage". This phrasing indicates that the piercing damage is the weapon's normal damage and the fire damage is "extra" damage added on top of that normal damage. The fact that the static modifier (which matches the devil's dexterity modifier) is applied to the piercing damage is further evidence of this conclusion. So, if the monk's dexterity modifier is +4 and their martial arts die is 1d6, then the damage of the spine when used by the monk in this way is 1d6 + 4 piercing damage plus 1d6 fire damage.
Drider longbow attack
Longbow. Ranged Weapon Attack: +6 to hit, range 150/600 ft., one target. Hit: 7 (1d8 + 3) piercing damage plus 4 (1d8) poison damage.
This attack has the same form as the tail spine attack, with the piercing damage being the "normal damage" and the poison damage being additional. The only difference here is that the normal damage die is 1d8, whereas our monk's martial arts die is a d6. That is no problem, however, since the monk is not required to use their martial arts die in place of the weapon's own damage die. Hence, the damage when the monk makes an attack with the drider's longbow arrow is 1d8 + 4 piercing damage plus 1d8 poison damage.
Sprite shortbow attack
Shortbow. Ranged Weapon Attack: +6 to hit, range 40/160 ft., one target. Hit: 1 piercing damage, and [additional non-damage effects]
Here, the confusion stems from the fact that there is no damage die. However, this is not a problem. Recall that a monk also uses their martial arts die to replace the normal damage of an unarmed strike, which is 1, just like the sprite's shortbow. If the monk can replace the 1 damage of an unarmed strike with their martial arts die, there's no reason the same shouldn't apply to any weapon with a normal damage of 1. Hence, the monk's attack with the sprite's shortbow arrow will deal 1d6 + 4 piercing damage. (Note that the monk could choose not to use their martial arts die and leave the damage at 1, but I believe their dexterity modifier would still be added to make it 5 damage total) In addition, nothing about the monk's Deflect Missiles ability takes away the additional effects of the attack, so the additional non-damage effects of the attack also apply to the target of the monk's attack (just like the additional damage effects in the previous 2 examples).
In summary, we had to parse a bit of English to figure out the "normal damage" of weapons with multiple damage rolls, and we used the example of unarmed strikes to conclude that the martial arts die can even replace the "normal damage" of a weapon with no damage roll at all. Once we did that, we simply applied the rules for Deflect Missiles and monk weapons in a straightforward manner to arrive at the result. In general, the missile, when used by the monk as a monk weapon, changes its damage die, attack and damage modifiers, and range, but it otherwise functions identically to the original attack.