What really is the self, and what really is a new soul?
Ah archival! Some big questions here! I shall do my best to share my perspective for what it's worth. I am actually struggling to write a book on this topic (among other things) and I have given it a bit of thought as it happens.
In my view, the 'self' is made up of a combination of the body, the mind, the soul and what I am calling the 'ego' - meaning that 'I am' sense of self which everyone has somewhere in there. Other people may use other terms for these things, but those are the words I chose, because I thought they were clear and simple.
All these different aspects make up the individual. The body is pretty straightforward and easy to understand. It exists in the material world, it is born, it ages, it dies. You can see it, touch it and so forth.
The mind is that bit which thinks and evaluates, learns, understands and is eternally curious. People who have experienced NDEs report that it continues at least for a reasonable time after death. They certainly report thinking and evaluating and considering their predicament. My view is that it is somehow independent of the body. It is not to be confused with the physical brain any more than feeling of emotion should be confused with the physical heart (although these organs are somehow involved - I don't believe they are the authors of these experiences - more like the receptors or translaters).
The soul is that which exists outside of time - that which is the 'magic' bit, the 'other' bit, the mystical element, the immortal bit. Other people call it consciousness, awareness, spirit, God - I don't really mind what you call it. I call it the soul.
The ego - well that is the bit that says, 'hang on - but I am me!!! I am not part of the universe. I'm an individual! I am not just my body. I am'. The ego is that sense of the individuality of the self distinct from everything else in the world - self-awareness and self-definition. Throughout the ages, even though I may have a different body, different attitudes, thoughts, emotions, gender, circumstances and so on I always have that feeling that 'I am'. I certainly 'recognise myself' throughout all those experiences - even if I (now) am a little shocked at my own behaviour (then) on occasion. My actions certainly seemed perfectly congruent to me then. The ego might be a product of the mind. I am still pondering on that whole bit!
All these elements are deeply entwined and highly interactive. Take one away, and everything changes.
Are the number of physical bodies the factor to the souls incarnate, or vice versa?
I think it is both. It is difficult to have one without the other, eh? It takes two to make an incarnated soul. If we start trying to make a cause and effect equation here, chickens, eggs, destiny and so on we will get into another tangle.
But, long story short, I have read in various books on the topic of "life between lives" regressions that people say the soul is kind of hanging about 'in heaven' waiting for a suitable body to become available and when the 'time is right', they are ready to incarnate and down they go into a new body (round about between 3-6 months into the pregnancy on average).