Do Jews believe in Jesus?
1) Did the Rambam assert that Jesus exists?
The one comment in Rambam that is quoted is the censored comment in Law of Kings and their Wars 11:10, preserved in the Karpach version (which was preserved in the Arab world, saving it from Christian censors). אף ישוע הנוצרי שדימה שיהיה משיח, ונהרג בבית דין--כבר נתנבא בו דנייאל, שנאמר "ובני פריצי עמך, יינשאו להעמיד חזון--ונכשלו" (דנייאל יא,יד). וכי יש מכשול גדול מזה: שכל הנביאים דיברו שהמשיח גואל ישראל ומושיעם, ומקבץ נדחיהם ומחזק מצוותן; וזה גרם לאבד ישראל בחרב, ולפזר שאריתם ולהשפילם, ולהחליף התורה, ולהטעות רוב העולם לעבוד אלוה מבלעדי ה'.
The first words are, "Even Jesus the Notzri , who seemed that he would be the Messiah, and was killed by a (presumably Jewish) court, was already prophesied by Daniel 11:14, as it says "offshoots of your people will rise up to support a vision, and they will stumble ..."
I am not compelled that the Rambam is saying that Jesus existed. In my humble opinion (as if!), he is saying that the character Jesus (real or invented) was held out to be the Messiah and, according to the story, was killed in a Jewish court, thus disproving their contentions. And Daniel prophesied that offshoots of the Jewish nation (i.e. Christians) would try to support him, but that they would stumble. I think he can still escape with his honor of Jesus did not exist. He is just trying to use common tradition of Jesus to prove that the Messiah does not need to do miracles, and that they are not impressive to certify a candidate.
According to the NT, Jesus was convicted in a Jewish court to death (what for is not clear), and turned over to the Romans for execution, which they did reluctantly on the insistence of the Jews. The Rambam says he was "killed by a Beit Din". According to some Jewish apocryphal sorces, Jesus was actually killed by the Beit Din. Rambam may have relied on these sources or may have been speaking imprecisely about the NT version. (Might he have read the NT? Oh my!) Either way, he feels the case that he was convicted by a Beit Din makes a much stronger statement against his Messianic claims, so he has a good reason to say it. Bottom line - no evidence that Rambam insists that Jesus actually exists.
2) Gemorra sources
I would look for the Gemorra to mention a historical character, between 10 BCE and 50 CE to have done something of lasting historical value, positive or negative, as an official "claim" that Jesus existed.
The Gemorra in Gittin 58 (learned on Tisha B'Av) mentions that three people were punished in heaven, Titus, Bilaam, and "sinners of Israel". Some hold that "sinners of Israel" was inserted by censors in place of You-kno'-'oo. However it is clear from the text that the Gemorra is giving a fanciful allegory about deserved punishment, not a historical narrative. It does not say what they or "he" did to deserve being boiled eternally in excrement.
The Gemorrah in Sanhedrin, perek haChelek (108 toward the bottom if I recall) relates a story of Yeshu HaNotzri (transliterated as Jesus) accompanying Rabbi Yehoshua ben Parachya to an inn. In the story, RYbP complements the inn, and Jesus thinks he means the innkeeper's wife, due to the similarity of the words. RYbP excommunicates him for occupying himself in such thoughts, both a false suspicion and revealing what is going on in the student's own thoughts, a huge Pharisee no-no. After shunning Jesus harshly several times, RYbP decides to forgive him, but Jesus approaches during prayer, and he mistakes RYbP's ignoring him for a final negative answer. Then, he goes to a public place in the Temple and worships an idol, as a sign that he will never be forgiven. RYbP exhorts him to repent, but he points out that RYbP himself has taught that there is no forgiveness. From this the Gemorra learns that one should not push away a student harshly.
Bottom line - This story would have taken place at least a century and a half too early to be the NT character. Even so, the Gemorra blames RYbP for the incident. I have often wondered if this Gemorra is not the source for the name, later attached to the legend. Rashi explains that Yeshu stands for "Yamach Shmo v'Zichro", may his name and mention be erased. Clearly the name is significant. HaNotzri means "the guardian" of the Torah. So, even though he was Tamudic scholar, and a Notri, he was nonetheless Yeshu, as his ego won out over his humility.
However, it cannot be a real 30 year old living in 30 CE or anything close if he was a student of RYbP. The fact that the Gemora does not blame him implies to me that it not equate with the NT character either.
There is one other famous mention in Sanhedrin (in the 40s, I think) which mentions that Yeshu HaNotzri had 5 students condemned in Beit Din. The names are not recognizable, except perhaps, Todah (Thadeus?). Each defendant quotes a verse with their name or something close implying that they should be saved, and the court quotes a verse with the name of each implying that they deserve death.
Bottom line - Whatever the underlying encoded message, it is clear that the Beit Din did not decide capital cases in this way. We do not hear of their real crimes. This story is cannot be literal. It is apocryphal.
3) Medrash - There are no contemporary medrashim that mention Jesus, that I am aware of. There are later ones, even a pamphlet called "Toldot Yeshu HaNotzri", likely dating to the late Middle Ages.
There is an interesting Notrikin in the Chumash, on the pasuk in parshat Va'Yechi (Gen 49:10).
לא-יסור שבט מיהודה, ומחקק מבין רגליו, עד כי-יבא שילה, ולו יקהת עמים
The roshei taivot spell out "lo yesh mum ra' k'yeshu", there is no evil blemish like Jesus.
Even if this is an authentic hint in a prescient Chumash, it can easily refer to the legendary Jesus rather than a particular man. Cool though, no?
Plenty of ink has spilled on the fact that there are no contemporary historical references to Jesus in Roman or Jewish sources. Josephus, especially, seems to have missed this very important revolution. In fact, there is one paragraph in Josephus that mentions matter of factly, that there was this guy Jesus and that the Jews condemned him and the R0mans killed him and that he was the Messiah. Then he continues the narrative of the story that he was telling in the previous paragraph. So other than this one paragraph, off the cuff, allusion to the greatest event in Jewish history, that every Jew has been awaiting since time immemorial, there seems to be nothing.
Bottom line - Even the Vatican no longer supports the authenticity of the Josephus reference.
So in summary, we really have no compelling case that Jesus was a historical figure. The concept that his name was Yeshua (Joshua) because he "saved the people from their sins" (Hebrew root yud, shin, ayin) does not mesh with the common transliteration IESVS, IE (yud), S (shin), V(vav), S (ending silent S in Roman). The name of Joshua the earlier biblical character is transliterated IESUAS, with the A standing for the voweled Ayin. In Toldot Yeshu, there is a reference to a student of the Rabbis that follows the Christians in order to remove the Jews from the movement. In the NT, Paul hears new prophecies from Jesus and encourages the movement away from Jewish practice, while insisting that Jews maintain the practice. Could he have used the earlier known tradition of Yeshu HaNotzri the idolator to create a code to drive the Jews apart from the non-Jews in the movement. The Jews might have recognized the reference and left the movement whereas the non-Jews would not. The movement then solidified all of its traditional characters under the name that Paul gave them. A grain of truth? Someone more ambitious than I can get a doctoral thesis out of that. But, at any rate, it must be clear that there is no evidence here to compel one to claim that Jewish sources insist that Jesus was a real person corresponding with the character of that name in the New Testament.