I started talking to Boddy before CH 1 ended, but finished before I leveled up again, so I didn't FINISH his quest. Will I get his next one?

As far as i know - No.

Sorry :(

You don't physically have to level up to get the next part of the story. In chapter 1, you need to be level 6 to get the whole story. There is a set level in each chapter. You do need to ask about the story three times, regardless of your level.

I believe that if you don't finish an NPCs quest fully in Chapter 1, you cannot continue to do the second one in Chapter 2. Right now, I am in Luskan in Ch2, and I finished Linu's 2nd quest because I had done her first one and gotten the necklace item from it...when I tried to initiate conversation to attain additional quests from the other NPCs in Ch2, they wouldn't advance to the point where a quest was given. So I guess you can't do Boddyknock's following quest if you are in Ch2 right now.

Yes, Debaser, you are correct. If you don't do the NPC quest in chapter 1, the later ones are unavailable. It's based off of the item received for the quest. If you lose the item, you also will not be able to do the later quests.