"Damage" isn't damage
If you have to put "damage" in scare quotes, it's not damage. Dipping a quill in ink doesn't damage ink, it just moves it about.
Conjuring a weapon
Using a minor conjuration dagger in combat is less certain, because "normal wear and tear" for a blade is nicks and chips, which is non-catastrophic damage, but is indeed damage instead of "damage" and will destroy a blade over time. Except that D&D doesn't normally care about wear and tear at all, even for weapons, and only cares about actual hit points of damage that structurally undermine objects.
D&D Designer Jeremy Crawford, rules guru for 5e, clarified in an unofficial tweet that using a minor conjuration weapon won't damage and dispel it:
Dario Berto @Uzedh · May 5
@JeremyECrawford If I conjure a weapon with Minor Conjuration and I hit a creature, does the weapon take a sort of damage to disappear?
Jeremy Crawford
@JeremyECrawford
@Uzedh No.
2:59 PM - 9 May 2016
But personally, I still think this falls into the realm of DM's judgement. Fifth Edition is designed around the principle that words mean what they mean ("natural language"), and that DMs are the local arbiters of the game's words so that they make sense for the game they're running. DMs are given, by the game's RAW, the responsibility and empowerment to figure out how to apply the meaning of words the game uses within the imaginary world.
This principle means it's possible that in one DM's world, deliberately or accidentally scratching a minor conjuration does dispel it. That DM's world is one in which conjurations are mystically fragile objects. In such a DM's world, Minor Conjuration wouldn't be useful for weapons, except perhaps if you were bluffing with them to appear armed when you're not. Another DM's world (Crawford's definitely!) might equally allow for cosmetic damage to be merely "damage" that doesn't count as actual damage. I expect most DMs are going to fall on the same side as Crawford and allow minor conjuration weapons to be useful.
The meaning of "object"
As for whether "an object" means a single object: yes, that's what the words normally mean. Since the ability's description doesn't include words that mean otherwise, Minor Conjuration is limited to creation "an object". Being listed for purchasing purposes in the equipment section doesn't have any significance beyond that you buy them as a bundle, so has no relevance for deciding the difference between an object and multiple objects.
As written, the bow creates its own ammunition "[w]hen you draw back its string and fire." As such, you cannot fire any other arrows you might happen to have on your person: the one the Moon Bow creates would get in the way.
That said, it would not be game breaking to allow the Moon Bow to fire magical arrows. Yes you will do extra damage but you are expending a magic item and a spell slot to do so.
Best Answer
I would say this is largely up to DM discretion. This can be seen 2 ways:
Because the item summoned is replicating an item that according to the PHB has specific damage values attached to it (The longbow specifically does 1d8 as per PHB pg 149 for example), I would think that yes, the bow/crossbow is dealing damage and would disappear.
Because the attack roll is for aiming an arrow, and the damage is piercing, from the arrow, the bow itself is not doing the damage. The bow/crossbow would only be doing damage if used as an improvised weapon to physically deal damage.
The intent of the errata ruling appears to have been to prevent the conjuration ability from being used to cause damage, however, when this exact topic was addressed by Jeremy Crawford here it was stated that a conjured weapon could be used.
To my knowledge this comment was written before the next printing of the book began, which contained the correction referenced in the errata.
Taking Jeremy Crawford's comment into consideration, I would again say that this falls to DM discretion, because 2 contradictory rulings have been given from official Wizards of the Coast sources. The most recent wording of this ruling would cause me to, however, believe that the former interpretation of the rule listed with the number 1 above, is the correct interpretation of the rule as of this point.