Was Stoning Still Officially Practiced?
As the previous answers have already said, no, it appears that stoning was abolished around the time of Jesus' birth.
Did It Still Happen Anyway?
Sometimes, but the best attested case took place under special circumstances.
The historian Josephus writes of an instance in which stonings occurred, probably around the year 62 CE. The short version is as follows:
The Roman prefect of Judæa, a man named Porcius Festus, died in office. This being the Ancient Roman Empire, it took a while for word of the death to reach the capital, and an equal amount of time for the new prefect, whose name was Albinus, to arrive in the region. The high priest at the time was Ananus, or Hanan. He had had problems with a few people in Jerusalem, including Jesus' brother, James. Josephus tells us that Ananus now exploited the temporary power vacuum, and had the alleged troublemakers stoned to death. The people of Jerusalem were outraged by this abuse of power, and immediately complained to Albinus, before he had even reached Judæa.
Here is Josephus' account:
CONCERNING ALBINUS UNDER WHOSE PROCURATORSHIP JAMES WAS SLAIN; AS ALSO WHAT EDIFICES WERE BUILT BY AGRIPPA.
AND now Caesar, upon hearing the death of Festus, sent Albinus into Judea, as procurator. But the king deprived Joseph of the high priesthood, and bestowed the succession to that dignity on the son of Ananus, who was also himself called Ananus. Now the report goes that this eldest Ananus proved a most fortunate man; for he had five sons who had all performed the office of a high priest to God, and who had himself enjoyed that dignity a long time formerly, which had never happened to any other of our high priests. But this younger Ananus, who, as we have told you already, took the high priesthood, was a bold man in his temper, and very insolent; he was also of the sect of the Sadducees, who are very rigid in judging offenders, above all the rest of the Jews, as we have already observed; when, therefore, Ananus was of this disposition, he thought he had now a proper opportunity [to exercise his authority]. Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrin of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others, and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned: but as for those who seemed the most equitable of the citizens, and such as were the most uneasy at the breach of the laws, they disliked what was done; they also sent to the king [Agrippa], desiring him to send to Ananus that he should act so no more, for that what he had already done was not to be justified; nay, some of them went also to meet Albinus, as he was upon his journey from Alexandria, and informed him that it was not lawful for Ananus to assemble a sanhedrin without his consent. Whereupon Albinus complied with what they said, and wrote in anger to Ananus, and threatened that he would bring him to punishment for what he had done; on which king Agrippa took the high priesthood from him, when he had ruled but three months, and made Jesus, the son of Damneus, high priest.
- Flavius Josephus, Antiquities of the Jews, Book XX, Chapter 9
Josephus explicitly states that it was illegal for the high priest to assemble a Sanhedrin without the consent of the prefect. If we assume that a stoning could not take place without the permission of a Sanhedrin, and if no Sanhedrin could be held without the consent of the prefect, it would appear that stonings could not take place without the approval, direct or indirect, of the Roman authorities. This agrees with the account provided by Josephus: it is plainly the case that Ananus wasn't deposed simply because he stoned someone; he was deposed because he stoned someone without the imprimatur of the Roman prefect. This makes it fairly clear that, regardless of whether stoning was still officially condoned by the Jewish authorities, it was still legally possible, but only if the Roman prefect had already given his consent.
This idea is further supported by the historical evidence:
The right of exercising capital punishment autonomously, even over their own countrymen, was withdrawn from the Jews by the Romans in the first... century [of the Common Era]. The precise date is controversial, but the limits are clear. E. R. Goodenough notes that the Greeks in Cyrene were allowed by Augustus, in a decree of 6 B.C., full judicial rights in everything short of the death penalty - this was reserved to the Roman governor, according to the customary provincial administrative pattern. Many scholars therefore maintain, with widespread Roman precedent, that the Jews lost the right of inflicting capital punishment in A.D. 6, when Palestine became a Roman province. Others believe that Jewish courts were allowed exceptional privilege in this matter until the Jewish revolt was crushed and the Temple destroyed in A.D. 70.
There seems no good reason to doubt either the intense dislike which the Rabbis felt towards death sentences under any circumstances, or their sincere desire to minimize the physical sufferings involved.
A man condemned to stoning was sometimes hurled over a precipice first, to abridge his suiferings. This was obviously the intention of the Nazarenes against Jesus in Luke 4: 29, whereas Stephen was judicially stoned on ground level in Acts 7. This certainly proves that such things happened under Roman rule by direct Jewish action - it neither proves not disproves their legality from a Roman viewpoint. John 8: 5 raises a question of principle, not of civic competence...
It would follow historical precedents if the Jews lost the ius gladii [literally, "the right of the sword", i.e., the legal right of a group of people to exercise capital punishment] in A.D. 6, when their land became a Roman province. Josephus describes the first Roman procurator Coponius as invested with the power of life and death by Caesar... But Josephus nowhere lays claim to any such Jewish authority in this period - indeed he blames the High Priest Ananus for overstepping his prerogatives by ordering on his own authority the stoning of James the brother of Jesus [and others].
The Jerusalem Talmud clearly states that Jews lost the power of capital punishment forty years before the destruction of the Herodian Temple, and the Babylonian Talmud echoes this1. Forty could be a round number, translatable into sixty-four. Newman believes that the Great Sanhedrin left the Hall of Hewn Stones in the Temple about A.D. 30, for reasons internal to Judaism, and that this fact, not Roman interference, caused the cessation of the death penalty. Roman practice does support the simple, factual understanding of John 18: 31: "We possess no civic power to impose a judicial death sentence at all." This was the interpretation of Schiirer, Mommsen, Bernard, Jeremias, Rosenblatt, and a host of other scholars.
Others insist that the Romans permitted [Jews] to exercise the death penalty, against Jews only and in matters exclusively religious, until A.D. 70. Jean Juster argues this with immense learning, scant respect for Gospel historicity, and a propensity to read into Josephus what suits him - that the High Priest's προστασία ["aegis", "auspices"] includes ius gladii is gratuitous assumption. T. A. Burkill reaches the same conclusion from the Temple inscription in Greek, warning Gentiles not to proceed beyond their court on pain of death2. Deissmann, however, attributes both inscription and penal procedure to Roman authority, which would invalidate Burkill's argument entirely.
The fact that many Jews were, like Stephen, judicially killed by their compatriots in many parts of the Roman Empire between A.D. 6 and 70 is not disputed. Goodenough demonstrates considerable laxity in the Alexandria of Philo, provided this lynch law was confined to Jews on religious charges who were not Roman citizens, and Origen reveals a similar situation much later.
Ulpian states that the power of Caesar's deputy may be either "pure" or "mixed." The first includes the right of inflicting the death penalty, the second stops short of this. This is further clarified by another passage, which assigns capital jurisdiction to those who rule over entire provinces. Ulpian makes it clear that with such rulers the power of death sentence is entirely personal, and under no circumstances transferable to another - yet responsible officials may not indiscriminately set at liberty accused men whose cases they cannot hear in person.
Allowing for abuses, it would seem that in her provinces Rome kept the ius gladii jealously within the hands of her appointed officials, regulating even their lawful use of it somewhat carefully. It seems almost inconceivable that the Sanhedrin alone could have executed [people] legally.
- Roy A. Stewart, Judicial Procedure in New Testament Times
Forty years before the destruction of the Temple, the Sanhedrin were exiled and took up residence in Hanuth. Whereon R. Isaac b. Abudimi said: This is to teach that they did not try cases of Kenas. 'Cases of Kenas!' Can you really think so! Say rather, They did not try capital charges.
[Note: V. A.Z. 8b on Deut. XVII, 10: And thou shalt do according to the tenor of the sentence which they shall declare unto thee, from that place; this implies that it is the place that conditions the authority of the Sanhedrin in respect of the death sentence. [J. Sanh. I, 1 has, 'the right to try capital cases was taken away from them, i.e., by the Romans. For a full discussion of the subject v. Juster. op. cit, II, 138ff.]]
"It is taught that more than forty years before the destruction of the Temple, capital punishment was removed from Israel”
2The Temple inscription mentioned above is a stone discovered by archaeologists in 1871 on the Temple Mount; it has been dated to between 23 BCE and 70 CE (the year of the Temple's destruction). It is written in Koine Greek, and reads:
ΕΝΤΟΣ ΤΟΥ ΠΕΡΙ ΤΟ ΙΕΡΟΝ ΤΡΥΦΑΚΤΟΥ ΚΑΙ ΠΕΡΙΒΟΛΟΥ ΟΣ Δ ΑΝ ΛΗΦΘΗ ΕΑΥΤΩΙ ΑΙΤΙΟΣ ΕΣΤΑΙ ΔΙΑ ΤΟ ΕΞΑΚΟΛΟΥΘΕΙΝ ΘΑΝΑΤΟΝ
"No foreigner may enter within the balustrade around the sanctuary and the enclosure. Whoever is caught, on himself shall he put blame for the death which will ensue."
In light of Stewart's argument above, this suggests that the even if the Romans allowed Jews to execute other Jews for religious offenses, the Roman authorities were also willing to execute gentiles for offenses committed against Jewish law.
Further support comes from Professor Lawrence Schiffman, the Judge Abraham Lieberman Professor of Hebrew and Judaic Studies at New York University and Director of the Global Institute for Advanced Research in Jewish Studies:
In fact, Roman law prohibited capital punishment at the hands of local courts such as those of the Jews. Capital punishment in any case had been made virtually impossible according Jewish law, which required that the two witnesses see each other, that the witnesses warn the perpetrator, etc. — all making it almost impossible that that Jews would have wanted to actually go through with an execution. Under Roman rule, Jews themselves, without [the Roman prefect], without the Romans, would never have been permitted to carry on capital punishment of anybody.
- Prof. Lawrence Schiffman, New York University
As Stewart and Schiffman have shown, it seems almost certain that the Jews of Judæa lost the right to conduct executions around the turn of the century, and that the Sanhedrin had, around the same time, voluntarily surrendered their authority to conduct executions. The only ways a person could be executed were to obtain the imprimatur of the Roman prefect (the legal method) or gather up a lynch mob and do it yourself (the illegal method).
Obviously, the passage from John which you have quoted above falls under the second category1: no authorities, whether Roman or Jewish, are involved in this sequence of events. Rather, a mob is attempting to stone this woman of their own volition. Obviously, this means that the question of legal authority to exercise capital punishment is irrelevant; unruly mobs are, by definition, unconcerned with legal niceties. As Stewart argues, it is widely agreed that such killings did take place, and the issue of legality is not relevant to the point. It should go without saying that people have always killed one another, despite the widespread existence of laws prohibiting extrajudicial executions.
Your specific questions:
- Was stoning still practiced in Jerusalem around 30-32 CE?
Probably not very often in Jerusalem itself, because the Romans had a significant presence in the city, and it was most likely illegal under Roman law by this point. Outside of Jerusalem, in the countryside of Judæa and Galilee, where the Roman presence was negligible or nonexistent, stonings definitely still happened from time to time, but it was illegal under Roman law from some time around the first decade of the first century CE. In most cases, these stonings were probably the work of mob justice, not a legitimate judicial process. It is very difficult to ascertain with any certainty, primarily because of the dearth of historical records of events in the rural areas of the region.
- When was the last time in history that stoning was carried out by a Jewish court in accordance with Jewish law?
It is difficult to say with any certainty. Was the stoning of James carried out in accordance with Jewish law? Maybe, but it was obviously illegal under Roman law. If we can take at face value the claim that the Sanhedrin and other Jewish authorities surrendered the right to try capital offenses of their own volition, and that this policy was not reversed at a later date, then the stoning of James was illegal under Roman law and Jewish law.
- Do Jewish sources discuss whether the Roman government would interfere in Jewish capital punishments, such as stoning?
As shown above, yes, the Romans absolutely interfered with the Jewish authorities' right to try capital offenses and conduct executions.
1 It isn't strictly in accordance with the intended scope of this site, so I will add this as a footnote: Among critical scholars, it is widely accepted that the Gospel of John is entirely unreliable and inaccurate. As such, the passage you have quoted is almost certainly a description of events that never took place. John is quite clearly a fairly rabid anti-Semite, and this passage, like many others in his gospel, is at least partially motivated by a desire to demonize the Jews. If you want an in depth explanation of why this is the case, it would probably be best to consult Biblical Hermeneutics.SE or a similar source for exegesis and historical criticism.