To answer as best as I can without spoiling things:
Reincarnation is well known. I'm fairly certain it's canon in Bleach, as it is, that souls move onto the path of reincarnation when they 'die' in the Soul Society. Or that's fanon that I've been exposed to enough to think it's canon. Either way, for the purposes of this, reincarnation as a concept is well-known. For someone to basically come back from the 'dead', as it were, is unheard of. There shouldn't be anything left, leave alone coming back basically intact into the person you reincarnated as.
Each of the choices directly shapes how much of 'Ichigo' is left in 'Kaien'. The more of Ichigo that's left, the more of his power and traits are left. The more influence it has on the character moving forward. It is 'Kaien' for the most part, regardless, but the choices decide how much is old and how much is new, if you will. In order:
Nobody: It's mostly Kaien. What's left of Ichigo is mostly memories and feelings towards his family (and friends), maybe a personality quirk or two.
Humanity: The middle ground, if you will. There's enough of Ichigo left that it certainly influences things a bit more, though again, still mostly Kaien. It's moved more to 'definite personality traits, and more raw feelings towards Ichigo's life'.
Two: The closest thing here to a 'personality mix'. Enough of Ichigo is left over that, to people who know him, it's still recognizably Ichigo. Kaien may be the core, but Ichigo remains there enough to really impact his personality and actions. If the character himself is Kaien, then this choice leaves enough of Ichigo around to make him more rough than Kaien ever was. Also expect more DRAMA in regards to the Kurosaki family.
(I'm not going to specify on the power side of things, though. That is spoiler territory)