Two of my friends have cleared up through every Forest singleplayer quest in an attempt to unlock the ability to convert enemy parts into weapons. However, they've cleared all the missions in the Forest now, and defeated the Dragon multiple times after the fact, and the next tiers of missions are not unlocking. Did something happen?

You need to complete the government quests in Multimode to progress through the solo mode quests.

This would mean you need to make a Multimode game and then do the quests 1-1, 1-2, etc... from Momoka/Nol in the Principal's Office / Lab.