Baam is an Irregular. Only Irregulars can make deals with floor guardians directly (like he did near the end of part 1). It's understood near the end of part 1 that Lahel made a deal beyond the standard test with Headon (rabbit-looking fellow, also floor 1 Guardian) but I can't remember if the details behind her deal are clear or not. She'd be an Irregular as well. (Really Rachel, but I liked the early use of Lahel better and prefer to keep using it). King Zahard was an Irregular. He made a pact with all the Guardians and has been allowed to have his rankers administer tests to allow regulars to climb the tower without dealing with the Guardians directly. Irregulars are the original tower climbers I believe, and when Zahard made the new rules it allowed Regulars (the chosen) to start the climb rather than just Irregulars (the strong willed).
The concept of Irregular vs Regular is not well understood (at least for me). Likely due to the majority of the detail being explained in the story/explanations offered outside the releases themselves. As I understand it a Regular is someone chosen by Zahard's people or Headon. An Irregular is someone not directly chosen but who's will is strong enough to access the tower (like the climbers before/during Zahard). The tower itself doesn't fear or hate Irregulars, and by extension neither does Headon, but many of the inhabitants and rankers do because of people like Zahard and the other known Irregulars. Headon says Baam is worthy, but I don't think that means he is not an Irregular.
The story has its mysteries, but I don't think denouncing them as a "mystery for the sake of mystery" is not very fair. I think it would have been kind of a boring story if they had started right when the gal asks Baam to join FUG and continued from there rather than skipping ahead like it did. And while it would be nice to see what happened to the original crew from part 1, it's understandable that with everyone thinking Baam died, that he'd be left behind while they continued the climb. They are now two separate stories. Whether or not the author will decide to touch upon the status of the first crew beyond references or specific characters remains to be seen. I think the two princesses (Anak and Androssi) and Koon will definitely make appearances due to their status in the families. And the masked woman who invites Baam to join FUG. While I would like Rak to come back and play with his Turtles (chibi-Rak made for some decent comic relief), the others are "meh" in my mind. Not that I didn't like any of them, just that they aren't important to the story.
And Baam must be a hotty in the Tower of God universe. Powerful women seem to fall in love with him quite easily. (Androssi, Anak even, Yuri to an extent, the female spirit inhabiting Black March, the new fire lady Yeon has a thing for him as well).
Can't sleep, clown will eat me.