... and the fact that, if it were a natural weapon, the monk wouldn't need Improved Unarmed Strike, and the interaction of a monk attacking with monk or non-monk weapons in conjunction with unarmed strikes, and the interaction of monk attacks with the TWF rules more generally, among others.
For 1, it's actually spelled out what appendages a monk uses. It's in the Combat section:
Etc. That's the reason a monk can wield a longspear and still attack with his unarmed strike as an light off-hand weapon.
"Light off-hand weapon" is a meaningless distinction when applied to natural weapons, which are either primary or secondary.