actually that is your progress, you have to start the quest pick your weapon then go fight your NM and get your double meld.... thats how the quest begins...
No the quest begins before that. You flag the quest when you choose which weapon you want to get. Then the quest appears in your journal.

We don't know if that step will stay the same. I predict that it will not. If I'm wrong, I'm wrong, but considering that everything else in that quest will be disappeared, I don't see why this won't change as well.