Learning gemara I did not see any death sentences that were carried out by the Bet Din. Did the Bet Din ever carry out a death penalty case?
The last Mishnah in the first chapter of Makkos states:
A SANHEDRIN THAT EFFECTS AN EXECUTION ONCE IN SEVEN YEARS, IS BRANDED A DESTRUCTIVE TRIBUNAL; R. ELIEZER B. AZARIAH SAYS: ONCE IN SEVENTY YEARS. R. TARFON AND R. AKIBA SAY: WERE WE MEMBERS OF A SANHEDRIN, NO PERSON WOULD EVER BE PUT TO DEATH. [THEREUPON] RABBAN SIMEON B. GAMALIEL REMARKED, [YEA] AND THEY WOULD ALSO MULTIPLY SHEDDERS OF BLOOD IN ISRAEL! (Soncino translation)
The Gemara (7a) adds:
A SANHEDRIN THAT EFFECTS AN EXECUTION ONCE IN SEVEN YEARS IS BRANDED A DESTRUCTIVE TRIBUNAL; R. ELIEZER B. AZARIAH SAYS, ONCE IN SEVENTY YEARS. The question was raised whether the comment [of R. Eliezer b. Azariah was a censure, namely] that even one death-sentence in seventy years branded the Sanhedrin as a destructive tribunal, or [a mere observation] that it ordinarily happened but once in seventy years? — It stands [undecided]. (Soncino translation)
The implication is that executions did happen, but very rarely.
In fact, just a couple of pages earlier (5b) the Gemara records a case of a false witness being put to death:
It has been taught: R. Judah b. Tabbai said: ‘May I [never] see consolation [of Israel] if I did not have one zomemim-witness done to death to disabuse the mind of the Sadducees, who used to say that zomemim [found guilty] were put to death only after the [falsely] accused person had [actually] been executed.’ Said Simeon b. Shetah to him: ‘May I [never] see consolation [of Israel] if you have not shed innocent blood because the Sages declared that witnesses found to be zomemim are not put to death until both have been proved as such, and are not [juridically] flogged until both have been proved as such.’ Forthwith did Judah b. Tabbai take upon himself a resolve never to deliver a decision save in the presence of Simeon b. Shetah.29 And all through his [remaining] days, Judah b. Tabbai used to go and prostrate himself on the grave of that [slain] witness, and his voice would be heard and people thought that it was the voice of the slain man; but he would tell them, ‘It is my voice! You will be convinced when on the morrow of this man's [his own] death his voice will be heard no more’. (Soncino translation)
Another example of execution occurs in Sanhedrin 52b.
Imarta the daughter of Tali, a priest, committed adultery. Thereupon R. Hama b. Tobiah had her surrounded by faggots and burnt. R. Joseph2 said: He [R. Hama] was ignorant of two laws. He was ignorant of R. Mathna's dictum3 and of the following Baraitha: And thou shalt come unto the priests, the Levites, and unto the judge that shall be in those days:4 This teaches that when the priesthood is functioning [in the Temple], the judge functions [in respect of capital punishment]; but when the priesthood is not functioning, the judge may not function.5 (Soncino translation)
However, in this case R. Solomon Luria suggests that it was an extra-judicial execution.
Also, the court of Moses executed the wood-gatherer (Numbers 15:32-36) and the blasphemer (Leviticus 24:10-23).
Yes. There are cases recorded in the gemara of Bet Din carrying out executions. Here is one example from Sanhedrin 45b:
Rabbi Eliezer said to the Rabbis: Did Shimon ben Shataḥ not hang in Ashkelon women who were found guilty of witchcraft?
(I am aware that the mishna goes on to point out that this may have been carried out in extraordinary circumstances, rather than following the strict protocols usually employed, but it is still an example of execution by Bet Din.)