I would say "maybe" or "it depends" to both.
- In Gittin 56b-57a there's a story in which Onkelos bar Kalonikos, the nephew of Titus, wishes to convert to Judaism, but before that, consults three evil men through necromancy. One of these is a man called "יש"ו"/"Jesus":
"Onkelos then went and raised Jesus the Nazarene1 from the grave through necromancy. Onkelos said to him: Who is most important in that world where you are now? Jesus said to him: The Jewish people. Onkelos asked him: Should I then attach myself to them in this world? Jesus said to him: Their welfare you shall seek, their misfortune you shall not seek, for anyone who touches them is regarded as if he were touching the apple of his eye (see Zechariah 2:12). Onkelos said to him: What is the punishment of that man, a euphemism for Jesus himself, in the next world? Jesus said to him: He is punished with boiling excrement. As the Master said: Anyone who mocks the words of the Sages will be sentenced to boiling excrement. And this was his sin, as he mocked the words of the Sages. The Gemara comments: Come and see the difference between the sinners of Israel and the prophets of the nations of the world. As Balaam, who was a prophet, wished Israel harm, whereas Jesus the Nazarene, who was a Jewish sinner, sought their well-being."
As you can see, there is indeed a story in the Talmud about a man called Jesus being burned in excrement. However, there are multiple stories in the Talmud in which a man called "Jesus" is mentioned, and not all stories perfectly line up together, and also not necessarily with the figure known as "the historical Jesus" (that they don't fit with the NT is of less consequence to Judaism). As such, there are those that believe that none of these stories refer to the same Jesus, like Rabbi Gil Student in the link that @TurkHill brought, or only a couple of them. There are those that disagree, however, and say that at least some of the stories refer to the same Jesus, and perhaps even all of the stories can be connected together.
For example, the Maharal of Prague wrote on this story in his book Netzach Yisrael. His version has some variants to our edition of the Talmud, most notably that in his version the Jesus mentioned in Gittin is not called "Jesus" but "Hahu Gavra", "that man", which is a long-in-use Jewish term for the famous Jesus. This is likely the result of censorship - the Church sought to censor all mentions of Jesus in the Talmud, on the basis of anti-Christian polemics. The result would be editing references to Jesus or even completely editing out whole portions of the Talmud.
- In Shabbat 104b it says:
"It was taught in a baraita that Rabbi Eliezer said to the Rabbis: Didn’t the infamous ben Stada take magic spells out of Egypt in a scratch on his flesh? They said to him: He was a fool, and you cannot cite proof from a fool. That is not the way that most people write. [An addition from the missing portions of the Talmud: Incidentally, the Gemara asks: Why did they call him ben Stada, when he was the son of Pandeira? Rav Ḥisda said: His mother’s husband, who acted as his father, was named Stada, but the one who had relations with his mother and fathered him was named Pandeira. The Gemara asks: Wasn’t his mother’s husband Pappos ben Yehuda? Rather, his mother was named Stada and he was named ben Stada after her. The Gemara asks: But wasn’t his mother Miriam, who braided women’s hair? The Gemara explains: That is not a contradiction. Rather, Stada was merely a nickname, as they say in Pumbedita: This one strayed [setat da] from her husband.]"
This story tells of this Miriam person who cheated on her husband, Pappos ben Yehuda, with a man named Pandeira, and this resulted in the birth of a man nicknamed "ben Stada", who brought witchcraft to Judea.
Like in the previous story, some think this story refers to Jesus. One reason is that in the Aramaic, "Miriam who braided women's hair" is מרים מגדלא שיער נשיא - "Miriam megadla se'ar neshia", which sounds an awful lot like Mary Magdalene.
Another reason is because of the version of this story that's brought in Sanhedrin 67a, which is one of the Talmudic sections that was clearly censored by the Church:
"[an addition from the missing portions of the Talmud: And the court did the same to an inciter named ben Setada, from the city of Lod, and they hanged him on Passover eve. The Gemara asks: Why did they call him ben Setada, when he was the son of Pandeira? Rav Ḥisda says: Perhaps his mother’s husband, who acted as his father, was named Setada, but his mother’s paramour, who fathered this mamzer, was named Pandeira. The Gemara challenges: But his mother’s husband was Pappos ben Yehuda, not Setada. Rather, perhaps his mother was named Setada, and he was named ben Setada after her. The Gemara challenges: But his mother was Miriam, who braided women’s hair. The Gemara explains: That is not a contradiction; Setada was merely a nickname, as they say in Pumbedita: This one strayed [setat da] from her husband.]"
As you can see, in this version, ben Stada was hanged on Passover Eve, which is another connection to the NT Jesus.
A third reason is a parallel story in the Talmud, in Sanhedrin 43a, which says:
"...from the outset, before the accused is convicted, he does not go out. The Gemara raises a difficulty: But isn’t it taught in a baraita: On Passover Eve they hung the corpse of Jesus the Nazarene2 after they killed him by way of stoning. And a crier went out before him for forty days, publicly proclaiming: Jesus [the Nazarene]1 is going out to be stoned because he practiced sorcery, incited people to idol worship, and led the Jewish people astray. Anyone who knows of a reason to acquit him should come forward and teach it on his behalf. And the court did not find a reason to acquit him, and so they stoned him and hung his corpse on Passover eve."
So we see here that was a man called Jesus who was also put to death on Passover Eve on the charge of, among other things, witchcraft, much like ben Stada. The Talmud in Sanhedrin then brings another story about the trial of five students of this Jesus: Mattai, Nakai, Netzer, Buni, and Toda.
Some have speculated parallels between these five and characters in the NT. Mattai is of course similar to Matthew. Netzer is similar to Nazarene (especially in Hebrew where they have the same root letters - נצר (Netzer)). Nakai might be Nicodemus. Buni might be Barnabas. Toda might be Thaddeus.
But like with the story in Gittin, there are those that disagree with this interpretation. They may say that this might be a figurative tale in the Talmud, or perhaps it tells of another such sectarian with the same name, or perhaps by the time the Talmud was written, the name "Jesus" was a generic name for "bad man", or that Mary Magdalene wasn't even Jesus's mother, and so forth.
To conclude, the Talmud is vague in these stories. Whether on purpose or not - we can't be sure. Who these stories refer to - that's open to debate, much like how it's being debated to this day how much of the NT is reliable (in particular considering contradictions between the Gospels).
1 So in MS. Vatican 130. Other textual witnesses in the Genizah website are either too badly preserved, simply have "ישו" or were censored and have "ההוא גברא" (that man) instead.
2 So in MSS. Munich 95, Harav Herzog, Florence 8-9 and Karlsruhe Reuchlin 2.
