The sovereignty of Edom/Idumea was destroyed, but its people remained and mostly converted to Judaism. The prophecy of Edom's destruction may have been fulfilled when Judean forces conquered Idumaea, or these prophecies may have been suspended when the Idumeans accepted Judaism (see excerpts from Jeremiah 12 and 18 below).
After the conquest of Judah by the Babylon, Edomites were settled in the region south of Hebron. They reportedly prospered there, and were given the the name "Idumaean" by the Greeks and Romans. The prophecies in the OP may have been fulfilled during the revolt of the Maccabees against the Seleucid kingdom. The Jewish patriot Judas Maccabeus conquered their territory for a time around 163 B.C.E. Idumea was again subdued by the independent Jewish ruler John Hyrcanus (c. 125 B.C.E.). It is not known how bloody these wars were, but Edom was not physically destroyed in the process. Instead, the Idumeans were forcibly converted to Judaism.
“Hyrcanus took also Dora and Marissa, cities of Idumea, and subdued
all the Idumeans; and permitted them to stay in that country, if they
would circumcise their genitals, and make use of the laws of the Jews;
and they were so desirous of living in the country of their
forefathers, that they submitted to the use of circumcision, and of
the rest of the Jewish ways of living; at which time therefore this
befell them, that they were hereafter no other than Jews.” (Josephus: Antiquities
of the Jews - Book XIII chap. 9)
Although generally looked down on by Jews, several Idumeans rose to great prominence, including Herod the Great and his son Antipas. - Source: New World Encyclopedia: Edom
The willingness of the Jews to assimilate the Edomites instead of destroying them according to prophecy may be understood in light of the following commandment:
Deuteronomy 23:7 - You shall not abhor any of the Edomites, for they
are your kin.
See also Jeremiah 12:
Thus says the Lord concerning all my evil neighbors who touch the
heritage that I have given my people Israel to inherit: I am about to
pluck them up from their land, and I will pluck up the house of Judah
from among them. And after I have plucked them up, I will again
have compassion on them, and I will bring them again to their heritage
and to their land, every one of them. And then, if they will
diligently learn the ways of my people... then they
shall be built up in the midst of my people.
This may be the prophecy that was fulfilled in the case of Edom. On the basis of Jer. 18:8 ("If that nation, against whom I have pronounced, turn from their evil, I will repent of the evil that I thought to do unto them") God's forgiveness in such cases superseded the need for prophecy to be fulfilled. (See also, the Book of Jonah.)
The Edomites have disappeared as a distinct people but many of them were known to have become Jews, and their descendants are likely still alive today.