You theorised correctly - it's been outright stated that this is the case. See: Nnoitra's recent Nell flashback.
I contend that souls bleed because they're humans made out of spiritons instead of atoms.
I also suggest that new souls originate the SS nobility rather than in the process of reincarnation. A person gets born into one of the noble families, lives out his/her life and is reborn in the human world with a memory wipe. From there, the person lives until, well, he/she dies and goes back to SS with memories (generally) intact. Then it loops round again, and the soul goes through the cycle, with detours into haunting, hollowdom and hell, until he/she gets eaten by a hollow, destroyed by a quincy, or otherwise terminated permenently.