From what I read, it sounded like it works a lot like Nyzul did in FFXI. That is, bosses will be every 10 floors and that's when you can save your data. For example. You'll start at floor one. Clearing floor 10 (so starting at 11) will be the first opportunity to save your floor progression data. You could only have floor progression data at floors 1, 11, 21, 31, and 41. Weapon and armor upgrades save immediately.

Also, yea since those Padjal weapons seem to be the only new weapons added, and no new armor I think it's safe to assume that that's what those are.