Conjurations are normally invincible unless stated otherwise
Yes, your Conjuration is invincible. Mostly*. According to the Compendium entry on Conjurations, a conjuration cannot be attacked unless the power says it can be, and Twilight Assassin doesn't say the conjuration can be attacked:
An effect type. A conjuration power produces a conjuration, which is a creation of magical energy that resembles a creature, an object, or some other phenomenon. Even if a conjuration looks like a creature or displays some degree of sentience, it is not considered a creature. Unless otherwise noted, a conjuration uses the following rules.
Creator’s Defenses: Normally, a conjuration cannot be attacked or physically affected. If a conjuration can be attacked or physically affected, it uses its creator’s defenses. Unless an attack specifically targets conjurations, only the attack’s damage (not including ongoing damage) affects the conjuration. For instance, an attack power that would cause a creature to take 20 cold damage and become immobilized would instead deal only the cold damage to a conjuration.
Contrast Twilight Assassin to the Shaped Consciousness psionic conjuration power, which explicitly allows its conjuration to be attacked:
Effect: You conjure a fragment of your consciousness in an unoccupied square in the burst. The fragment lasts until the end of the encounter or until you dismiss it as a minor action. The fragment occupies its square, and you and your allies can move through it. When you take a move action, you can also move your fragment a number of squares equal to your speed. You can see, hear, and use psion powers as if you were in your fragment’s space.
Your fragment can be attacked. If it takes any damage, it disappears. It is immune to your attacks.
* Here's the part of the conjuration rules that makes your conjuration not 100% invincible, though: it only survives as long as you do. :)
Death Ends: A conjuration ends immediately if its creator dies.
Attacking with a Conjuration
If you crit with the conjuration, does it really use whatever crit bonuses you would get from the item (like a magic Ki-Focus or magic weapon)?
Yes. The rules on conjurations attacking state the following:
Attacking with a Conjuration: Normally, a conjuration cannot attack. If a conjuration can attack, its creator makes the attack, determining line of effect from the conjuration as if the creator were in the conjuration’s space. If line of sight is relevant, determine it from the creator, not the conjuration.
Since it is technically you using the Twilight Assassin Attack power for it, all gear you're wearing that would affect that power applies. It's an implement power, so your ki focus applies too.