So here is Togen's short form for those questions. I know its not as obscure as zen is supposed to be, but I can't always be as cosmic as I should be.
First off Buddhist don't subscribe to reincarnation. reincarnation is the transmigration of the soul from one life time to the next. We believe in rebirth. This is rebirth without a transmigrating eternal, unchanging intrinsically existing soul or self.
So you ask what is going on here? The best thing to do here I think is to look at how Buddha explained our identity in this single life time. Buddha said , what we are, how we really exist is as a functionally unified combination of five aggregates. These five aggregates fall into two groups or types of processes.
First there is a material process, our body, which can be seen as a current of material energy. Then there is a mental process , a current of cognitive happenings. Discrete events of cognition. Both these currents consist of factors that are subject to momentary arising and passing away. The mind is a series of mental acts made up of feelings, perceptions, mental formations and consciousness. I suggest you look at my blog on mind.
In Pali these mental acts are called "cittas". Each citta arises, breaks up and passes away. When it breaks up it dose not leave any traces behind. It dose not have any core or inner essence that remains or is passed on. But as soon as this mind or citta breaks up, immediately afterwards there arises another citta. So we have a succession of cittas, or a chain of momentary acts of consciousness.
When each citta or mind falls away it transmits to its successor whatever impressions have been recorded on itself, whatever experiences it has undergone. Its perceptions, emotions and volitional force are passed on to the next citta. Imagine one pool ball on a pool table striking another and passing on its speed, direction and momentum. Each of these emotions and the very volitional force or our intentions is passed on to the next citta and in that way all the experiences we undergo leave their imprint on the onward flow of consciousness which is called the "cittasantana" or the continuum of mind.
This transmission of all our experiences and intentional actions , this causal continuity, gives us our continued identity. We remain the same person yet not the same person throughout our lifetime. From moment to moment and from life time to life time we carry the imprints of the cittas that proceeded us. So there is no self, but you have a valid sense of "I" so don't think you don't exist, you simply don't exist in the way you thought you did.
The traditional Buddhist believes that rebirth takes place through out six realms of existence. These we call the desire realms.
The god domain
The jealous god domain
The human domain
The animal domain
The hungry ghost domain
The hell domain
All these realms are just associated with this one universe; they also believe that there are countless universes in existence at any one time. So the current population of this human realm on this one planet is and has been just a small percentage of the sentient beings in existence.
Should the population here reach a hundred billion humans it would not begin to reach the countless population of all sentient beings.
I hate defiling the concepts here, I am sure Dogen would say I am just passing along a mundane view of a inexpressible truth. But we gotta start some where.