I dunno if you should pay too much attention to what I say about reincarnation; I am not an expert, and can only tell you what I think based on my somewhat unique background.  I know very little about Pythagoras except his theorem and that he started a mystical Greek philosophical school.

There is a story in the Buddhist tradition of a Prince who was very much into the hunt, and spent all his time hunting and killing animals, and which the Buddha indicated, because of the Prince's desires, would probably be reborn a tiger.  Interestingly he is reported to not have said so in so many words but just with gestures.  The Buddha was not much into animal rebirths -- this is much more a Hindu idea, and I have no idea whatever what the Pythagoreans thought.

Pretty much everyone around here, including the Communists, "believes" in reincarnation, in the sense that the idea is part of the furniture and just taken for granted.  When you drill down, however, one finds differing theories and some skepticism.

Personally I am very skeptical, and get more and more so as I age.

First one knows that the teaching originated in India, probably as far back as the Harrappa cultures (sometimes spelled with just one "r").  Unlike many of the features of modern Hinduism, such as casts and the panoply of deities, the teaching of reincarnation is common to all the religious traditions originating in India, and hence one can conclude it was probably there before the Aryan invasions (which brought in the Indo-European deities).

This is not to say the idea that we are reborn (the technically correct term -- "reincarnation" has to do with coming back to earth as a new incarnation, not as a new person) has not appeared elsewhere.  It is found as a speculation in Greek philosophy in such people as Plato, but was not as far as I know a belief in Greek religion.  Some assert that the Greeks got the idea from India, but records of such Greek thinking predate Alexander, so I doubt it -- more likely the Greek philosophers thought it up on their own.

As found in popular thinking nowadays, both in India and in the Buddhist areas, the teaching can be seen to have two components -- belief in karma and belief in the immortal soul.  Karma, as we all know, is the tendency to accumulate brownie points for good behavior and bad marks for bad behavior in a cosmic account.  Expressed in a more sophisticated way, what it means is that our behavior changes us (our soul) as we live, for the good and for the bad, and this accumulates and eventually, as a part of the universe's cause and effect processes, leads to good or bad "luck."   They also lead to rebirth in a better or worse position than one's present life.

What is seen to happen when one dies is the body is gone, but the soul or spirit persists, and desperately desire ("grasps") to get back what it just lost -- sensations mainly (hence the "hungry ghost").  The positive karma one has gives one somehow a certain power to influence one's rebirth (which is pretty much the only way to escape the disembodied -- ghostly -- state) -- to something desirable (maybe a heaven) or something undesirable (maybe a hell) but more likely to another human -- again to high or low state.  The widespread notion of being reborn an animal is also considered possible, but not likely since once having been a human there are very few who have any desire at all along these lines, and it is our desire that motivates the entire process (sentient animals are, however, seen as a virtually inexhaustible source of new human souls).

Unlike the Hindu approach, where this is seen as immortality, the Buddha saw it as a horrible trap -- called Samsara -- where one is trapped in a cycle of suffering -- dying is suffering, being born is suffering, life is suffering (I tend to see life, even the painful parts, differently).  Therefore he is reported to have discovered the way out -- through enlightenment, which is achieved through a life of meditation and goodness (basically by becoming a Buddhist monk -- which doesn't guarantee enlightenment but which is pretty much necessary).  Most Buddhists would like instead to go to one of the Buddhist heavens.

Enlightenment nowadays is seen as peace of mind, happiness, understanding of the human condition, and things like that, but the early conception was a supernatural awareness and level of knowledge that included memory of past lives.  In such a state one can die at any time and enter Nirvana -- often rendered "blessedness." Some see it as personal extinction, or maybe something like the Transcendentalist concept of rejoining the cosmic ocean of mind, or maybe just entering a special heaven where one's karma doesn't eventually run out.  Portrayals of it for children usually depict a (to me boring) state of people sitting around meditating.

Now some problems.  First, there is the convincing and I think very true Buddhist teaching of "no self." All of the above picture sees us as having souls, or selves, but Buddhist teaching from the very earliest tells us that this is an illusion.  Even mindful meditation can persuade those who know to look for it that there is no soul, no self, but only a process not unlike a burning candle or a wave on the water, or an electro-magnetic particle/wave duality we call a photon.

Then there is the modern scientific understanding that what we call mind is really just brain activity.  This concept has its own problem (the source of what are called "qualia" or sensations and emotions as we experience them -- the brain doubtlessly produces them but what is it that experiences what the brain generates).  Still, that problem aside, how does all this survive the destruction of brain (we see that mind tend to disappear even while the brain lives if it has some degenerative disease).  Where does the mind go -- into space as a disembodied ghost? -- the possible mechanism is an absurdity.

The general view in Vietnam is that there are ghosts everywhere -- spirits so evil they have not been able to be reborn (although why not one of the hells?).  These are frightening and malevolent and not even mentioned out loud.  There are also good spirits wandering around who for some reason haven't been reborn -- shrines can be seen here and there for these, to give them comfort.  Ghosts tend to be restricted to where they can travel at normal human speeds, so, except maybe for Pure Land Buddhists, we are reborn on the earth and the ghosts on the earth are those of dead humans.

One problem is expressed in the question, "Where is Mozart." The observation is that each of the world's great artists -- composers, authors, painters, etc., is noticeable for a clear "voice" that (at least in Mozart's case) is readily recognized from the very beginning.  Even though we are told we forget past lives, if the spirit persists, one would think that Mozart would by now have been reborn several times, further enriching our musical heritage, but we see nothing recognizable as such.

My main problem though is simply lack of evidence.  There is no credible evidence for ghosts, nor for past memories.  Unfortunately study of such phenomena, if there is any reality there, has been contaminated by air-heads and frauds and parents eager to make a lot of money from their precocious children by feeding them stuff.  As a result, I don't think it is possible for there to be an objective study, and serious scientists simply don't.  The few who do and who get results are soon denounced and debunked anyway, maybe properly.

There is also déjà-vu. There are efforts to "explain" this wide-spread phenomenon, and I think most of the explanations are valid most of the time, but it is impossible to rule out the possibility that a residue really are memories of past lives.