What is the source of Gilgulim (reincarnations) in Judaism? Are there people who disagree with its existence? Who was the first to mention it? What I am asking essentially is what is the historical development of Gilgulim in Judaism?
In regards to the first one, Gilgulim are first mentioned in the Heikhalot texts, also found in the Zohar, and Sefer HaBahir. According to those who hold the Zohar is Tannaic, that puts the idea at least as far back as Tannaim and Amoraim. Several of the Geonim argued over it, such as Sa'adia Gaon.
There is a piece in the Tshuvas HaRashba in siman 418 which denigrates the belief in gilgulim. It is a response from the Chachmei Luniel to the Rashba, proving that they don't believe in any nonjewish ideas. See there ד.ה. ומתועלותיה בסוד הנפש. One of the points they raise why it is an unjewish belief is because judgement and punishment is a fundamental belief in this religion, and if an evil man can come back as a righteous one and vice versa, the entire system doesn't work. Although this is not the Rashba talking, he did choose to put it in his tshuvos, seemingly accepting their argument of being good Jews. This is especially interesting in light of the fact that people like to quote the Ramban, the Rashba's teacher, as believing in gilgulim.
This is a list of opponents to the belief in gilgulim taken from here (2001): The Dove of War
As is well known, there have been many rabbinic authorities who subscribed to belief in gilgulim.On the other hand, there have also been numerous opponents to this belief, including Rav Saadiah Gaon (Emunos v’Dayos 6:8); Rabbeinu Avraham ben HaRambam (see R. Margoliyos, in his introduction to Milchamos Hashem p. 19 note 11); Rabbi Avraham ibn Daud (Emunah Ramah 7); Rabbeinu Yitzchak ben Avraham Ibn Latif (Rav Poalim, p. 9 section 21); Rav Chasdai Crescas (Ohr Hashem, ma’amar 4, derash 7); Rav Yosef Albo (Sefer HaIkkarim 4:29); and Rav Avraham Bedersi (Ktav Hitnatzlut leRashba). See too Rashash to Bava Metzia 107a (I am told that certain Chassidim will never study Rashash because of his comments on this topic). Also see Rabbi Samson Raphael Hirsch, commentary to Genesis 50:2. For further discussion, see Rabbi Yitzchak Blau, “Body And Soul: Tehiyyat ha-Metim and Gilgulim in Medieval and Modern Philosophy,” The Torah u-Madda Journal vol. 10
See there also for the bird story.
Rabbi Hirsch's words are particularly harsh describing what we call 'gilgulim' where "the soul did not remain in its personal individuality, but wandered from body to body-even to animals- in manifold metamorphosis" as a decidedly Egyptian belief in diametric contrast to Jewish ideas.
Rav Sadia Gaon also famously decried this belief as an impossible insult to the human soul, where we would find the human soul in the body of an animal.
If I'm not mistaken (if I recall correctly from an "intro to Kabbalistic concepts" lecture by Rabbi Breitowitz given at Ohr Sameyach; contact them for the audio), the notion of Gilgul (whereby a departed soul returns to earth in a different body) appears in a major way with the kabbalistic teachings of the Arizal in the 1500s. As we have none of the Arizal's original writings other than a receipt for a sum of pepper (he was a spice-seller), I suppose you'd have to consult his students' writings, such as those of R' Chayim Vital.
Does the concept appear in the Zohar? I don't know.
Certainly if you look through Sefer Chafetz Chaim and Shmiras HaLashon (c. 1900), while the author is known as a Halachist and not Kabbalist, he accepts this kabbalistic notion (and many others) and cites it vis-a-vis a punishment for speaking Lashon Hara ("lakelev tashlichun oso").
This somewhat contradicts those who believe in the notion of Gilgul.
Or in Yeshivish:
It's a shtickel of a shlug-up for the oilam that holds by the inyan of Gilgul.
I've heard that some Hassidim purposely avoid the Rashash's commentary in general, because of this.
Today it's certainly accepted in many circles; most siddurim have a version of Bedtime Prayers that forgives "all those who have wronged me, whether in this incarnation (gilgul) or another one." On the other hand, it's not one of the Thirteen Principles, and I know one rabbi (who I personally respect) who omits that phrase (not that he necessarily rejects Gilgul, just he's not sure how this forgiving-acts-from-other-incarnation works).
The Sefer Gilyon Ari (p. 40 - found on Otzar Hachochma) mentions the Rashash noted by Shalom. He adds that only a few rishonim questioned the concept of reincarnation and that it is widely accepted in traditional Jewish sources. Gilyon Ari then resolves the Rashash's difficulty by explaining the gemara in Bava Metzia (107a) as a reference to a person's first reincarnation:
וכן העיר ברש״ש כאן, וז״ל "דמכאן סתירה קצת לבעלי דעת הגלגול", ובאמת הגם שמקצת מהראשונים פקפקו בכללא דגלגולים, מ״מ רובם אשרוהו וקיימוהו כמש״כ בתשובת מהר״ל בן חביב סי׳ ח׳,ובפירוש תרגום יונתן בן עוזיאל (דברים לג-ו), ובספר שער הגלגולים לר״ח ויטל, וברמח״ל ספר המאמרים מאמר החכמה, ורבינו הגר״א משלי כא-טז, ויונה ד-ג. וע״ע באבן שלמה (פ״ג אות ה׳) אשר בזה מתבאר ענין צדיק ורע לו רשע וטוב לו, לפי שנגזר עליו מתחילת יצירתו מזל הראוי לפי מעשיו בגלגול ראשון, ע״ש. ולשיטתם צ״ל דסוגין איירי בגלגול ראשון דביאתו בלא חטא.
See: Sefer Bahir (as already mentioned), Reshit Hochma (Shaar HaYira 13),Ramhal (Derech Hasgem), Shaar HaGilgulim 22), Sefer Haredim (7:57), Degel Mahane Efraim (Parashat Mishpatim), Abarbanel (Debarim 25), Rabenu Bahya (Debarim 22:1), Ohr Hashem (4:7), Sefer HaIkarim 4:29). All these are sources for the discussion. EDIT: There is scientific proof to Gilgulim, and most of the Hachamim agree that Gilgulim exist.
See the Ralbach in his responsa 8 where he brings the two sides together, he then says that the Rasa"g and the like only had chochmah chitzonis i.e philosophy and the chacmei haemes ramban and others have stated that there are gilgulim - therefore we have to beleive in gilgulim. see also magen vetzinah (perek 12 and thirteen) from reb yitzchak eizek chaver a sefer written as a very sharp response response to reb yehuda aryeh di modina - he writes there that reb saddia never disputed gilgul only a certain type which the indians beleived in and when reb sadia says the belief of the yehudim in gilgul it means indians hodiim because he wouldnt refer to jews as other people the yehudim. the maharalbach says however not to talk about gilgulim in public as we see the rishonim and gemara only hinted to it and we can be no better then them! it seems that this has changed since toras harizal has become known to the massess - however i found an ohr sameach in hilchos yediah ubechirah who writes he will not disscuss gilgulim as the maharalbach has written not to and we keep this part of torah with talmidei chachmim tzenuim.