In case you're interested the way I see it is that something is born -- something physical, biological -- e.g. starting at conception with sperm and ovum etc.
That causes a blastocyst and so on -- which I doubt is a "sentient" being yet, though what do I know?
After (physical) birth the infant develops (psychologically), maybe acquires a sense of "Mum" and "Me" and "Dad" and "Food" and "Cold" and "wet" and so on and so on.
I suspect that ability to distinguish "self" results of acting semi-autonomously (e.g. "if 'I' start to cry, then Mum arrives") -- eventually the whole DO cycle, e.g. "I've learned I prefer the feelings associated with Mum being here, maybe feelings of needs (and eventually "cravings") satisfied, therefore I want Mum to arrive, therefore I intend to cry".
People (on this site) interpret "birth" and bhava and so in different ways, but I think that this (above) is at least one or two of those ways (i.e. physical birth plus the arising of a self-view or conceit, plus kamma as an intentional action, resulting from previously acquired/stored/interpreted/associative experience).
This is further complicated by two factors:
- This answer associates a self (i.e. "I" or "we") e.g. with physical form, a body (which anatta may suggest is a wrong view and/or well worth unlearning)
- There's more to a person (and a person receives more from their parents) than just DNA -- also food, a formal education, all informal experiences -- and not only from "parents" but from teachers, friends, the world -- and from "themselves" i.e. people are shaped by "their own intentions/actions"
Why were we physically born?
That's biology, isn't it? So e.g. "ask your Mum and Dad" about that?
Or do you mean, "why do 'we' associate 'me' with 'this body' or 'our body'"? In which case, I don't know, I think there was a metaphor in a sutta of "a dog, tied to a stake"...
Was it our personal kamma? Did we personally choose this?
I think it's better not to blame other people -- e.g. "you made me like this, you made me do this, it's your fault that I exist" and so on.
Does kamma create the self (person)? Or does self (the person) create kamma?
So far as I know, the text says that "people inherit their own kamma".
Also I think the Dependent Origination is taught as a (repeating) cycle, a.k.a. "the wheel of life".
Did the Buddha teach it cannot be determined whether the chicken comes before the egg or the egg comes before the chicken?
It's "a beginning is not evident", isn't it?
Is this why many Buddhists are vegetarians?
Well ... indirectly? Perhaps?
There are personal reasons, but I don't know that I can attribute those (project them onto) "many Buddhists".
That kind of doctrine (vegetarian) is more a feature of some schools than others.
I think the Buddha in the suttas says more about e.g. "gratitude and parents" than about "chickens and eggs" (though "not killing" and "metta" and so on is also, I think, clearly mentioned).