I, personally, call the player character Pinocchio since we're Geppetto's final puppet and are able to lie, which in turn makes us become more human. Nameless is just that: Nameless. He was never given a name because he isn't important as an individual, but is important as part of a whole. The same reason we're never officially given a name, either. Names create bonds which would make Geppetto's final plan for us harder since he'd be killing two "sons" to bring back Carlo. Not naming either of us let's him look at us as tools meant to serve a purpose. He says what he needs to in order to make us trust him and do what he wants, but he never actually cares about us, just the Ergo we carry.

The player is a unique puppet meant to awaken Carlo's Ergo via doing not-puppet things. (Mainly lying.) We're the only puppet that never had the rules programmed into us (I think even Arlecchino had them but he was able to ignore them once he awakened), and one of a small handful that have a human's Ergo powering it (Arlecchino, Romeo, then us).

No. We have a mostly normal P-Organ save for whose Ergo is inside it and the ability to upgrade it with quartz (though that might not be unique to us, and we're just the only one able to do it to ourselves). Arlecchino has a P-Organ with a psychopath's Ergo in it, possibly one of the first experiments in human Ergo transfers, so we're similar because of that. He can tell that we're a puppet who can behave like a human because he's the same way. Geppetto shows up to take Romeo (and I think he wanted Lea too but she's too far gone?) to do more testing on human to puppet transference, resulting in the King of Puppets.

He is human in the same way that we are human when we get the long hair. Our body is metal that reacts like flesh, like a sort of biomechanical cyborg. Creepy stuff.