weapon power that has an
effect that deals damage clearly falls under the wording of Frost Weapon:
All damage dealt by this weapon is cold damage.
And on vulnerability,
A creature that is vulnerable to a specified damage type usually takes a specific amount of extra damage when it takes damage of that type, or it suffers a specific effect. For example, a creature that has vulnerable 10 radiant takes 10 extra radiant damage when an attack deals radiant damage to it or when it takes ongoing radiant damage.
Vulnerability doesn't say anything about a power having to hit the target, just that it needs to take damage of that type. It's also pretty clear that vulnerability affects effects because the rules specifically mention ongoing damage, which is exclusively an effect.