First, if people are reborn again and again, why is the population expanding? Since the Bible only vaguely refers to reincarnation (Matthew 11.13-14, and there probably as a metaphor), we will have to use the teachings of those faiths that embrace reincarnation as well as reason for our answers. This first question underscores the difficulty in harmonizing the various reincarnation beliefs. It seems that most folk who believe in reincarnation think they come back as another human (Shirley MacClaine, et al). However, in most reincarnation systems, this isn't the case at all. In Hinduism and other Eastern faiths, when one has failed to reach their potential in their current life, they return to try again in a new form, and often as a new species (which gives rise to vegetarianism because you don't want to accidently consume your Aunt Beth). However, to answer the question, why would the human population be increasing? It would seem that since many animals and insects are dying each day (for instance, check your windshield, the meat counter at your grocery store, and the endangered species list), and since they too would be reborn into this world if they hadn't lived up to expectations, it is inevitable that the human population would increase.
Question two: if reincarnation is supposed to give us another chance to make good, why don't we remember our past lives? It would be helpful to remember our past lives so we could avoid repeating mistakes. However, in one of Jesus' parables the rich man in hades asked Abraham to send Lazarus from Paradise to earth to tell others about the fate of the opulent rich. But Abraham replies, "They have Moses and the prophets. They should listen to them. . . . If they do not listen to Moses and the prophets, neither will they be convinced even if someone rises from the dead" (Luke 16.29, 31). In other words, we wouldn't listen to ourselves anyway, so why bother? And besides, what could we learn as a human from a former life as an animal (except perhaps to be a kinder, gentler human, and not to exploit animals for food or clothing or shoes or. . . .)?
And finally, the family environment has a great influence on a person that could set them back on the road of enlightenment -- and is this fair? My own question is, is any of this fair? The concept of reincarnation is based on a capricious God who is waiting for us to finally "get it right" through study, deeds, or devotion. Accordingly very few -- very, very few -- ever achieve enlightenment and pass on from this world to the next. There is no room for error, no place for forgiveness, and so most of us are stuck in an endless cycle of trying to "get it right." But can we ever get it right?
Either way, it is true the family has a great deal of influence on us in every aspect of life. Who we become is largely based on who we've been and how we've been raised. But ultimately the responsibility for our actions cannot be shifted to our family. We are individually responsible for who we are and what we do -- family or not, reincarnation or not. Neither the God of Abraham nor the God of reincarnation cuts us any slack when it comes to excuses. The difference is that the God of Abraham, Isaac, and Jesus offers forgiveness and welcomes us as member of the family of God, while the God of reincarnation offers us another chance. And another. And another. And another.