There is no such general belief. There are instances when certain inter-caste marriages were deliberately conducted for a child to be born with specific qualities. Such ‘hybrids’ were possible only if the parents were ‘pure’. So the purity of the genes was advised so that the lineage of thinkers, rulers, traders and workers could be maintained. Today we maintain such purity for dogs and other pets, earlier human beings were given more importance.
Because He is Omnipotent and Omnipresent, He CAN take avatar, and so He takes!
Strictly speaking, Vedanta questions even the life after ‘birth’! From the stand point of the Supreme Reality, this life and the world itself does not have Absolute Reality.
However, at the transactional level, the journey of the individual self (the jīva) continues after the death of the body. That way, yes, there is life after death.
Knowing one’s own Self as the Self in all, and abiding in this knowledge itself is liberation.
To rid oneself of the burden of what one is not, and to abide in what one really is. The state of Supreme Bliss and Absolute Awareness is called liberation.