1) No, because as you've noted, it's a separate attack, and it doesn't have either an implement or a weapon keyword, so your implement or weapon isn't going to affect it.
2) Yes, the vulnerability would be pinged twice if you find a way to make both of them cold.
There are a couple of possible ways to do so, but they all bring up further rules questions:
1) Multi-class to Swordmage and take the Malec-Keth Janissary Paragon Path. At level 16, you deal 1d4 extra damage of a type, and cold is one of the choices. (Note that this opens up a rules-as-written can of worm over whether or not this attack now has a damage roll and gets additional bonuses).
2) Use the Blistering Flourish attack technique. The Rules-as-written controversy here would be over whether or not the text of:
Until the end of your next turn, your melee attacks deal extra fire damage equal to your Charisma modifier.
is converted by a Frost Weapon to Cold (this is tied up in the Adding/Removing damage types rules)-
(RC115)- Adding and Removing Damage Types: If a power gains or loses damage types, the power gains the keywords for any damage types that are added and it loses the keywords for any damage types that are removed.
So the question is whether or not Blistering Flourish's fire there is overwritten by "Cold."
One final note: be aware that the Theme power Sarifal's Blessing, from Sarifal Feywarden, has the potential to add elemental vulnerability without Lasting Frost for a few turns. It works more easily with Fire on a monk, though, than with Cold.