Short answer
The Hebrew has no unicorns. The King James Version took the idea from older Greek and Latin Bibles. It has "unicorn" nine times, where the Hebrew word, re'em, means a wild ox. Most of its "dragons" are jackals, serpents or great sea creatures in the Hebrew, and its "cockatrice" is a viper. The old Greek translation turned the wild ox into a "one-horned" beast, and the Latin often followed. Modern Bibles have replaced almost all of these creatures. The NIV keeps the dragon only in Revelation, where it stands for the devil. So for centuries, English readers took as God's (﷾) word creatures that the Hebrew never mentions. Islam's answer is the Qur'an, which God (﷾) Himself promised to guard (15:9). Muslims recite it in the Arabic in which it came down.
Full explanation
Unicorns in the King James Version
The word "unicorn" appears nine times in the King James Version. It is found in Numbers, Deuteronomy, Job, the Psalms and Isaiah. God's (﷾) strength is compared to a unicorn's, and Job is asked whether he can tame one.
“God brought them out of Egypt; he hath as it were the strength of an unicorn.”
“Will the unicorn be willing to serve thee, or abide by thy crib? Canst thou bind the unicorn with his band in the furrow? or will he harrow the valleys after thee?”
“And the unicorns shall come down with them, and the bullocks with the bulls; and their land shall be soaked with blood, and their dust made fat with fatness.”
What the Hebrew says: a wild ox
The Hebrew word behind every one of these unicorns is re'em, the wild ox. The Jewish Publication Society's translation of 1917 and the World English Bible both translate it that way. In Isaiah the "unicorns" come down together with bullocks and bulls. That fits cattle, not a horse with a horn.
Gives re'em as the wild ox; tan (plural tannim) as the jackal; tannin as a serpent, dragon or sea monster; and tsepha / tsiph'oni as a kind of poisonous snake.
“God brings them out of Egypt. He has as it were the strength of the wild ox.”
“And the wild-oxen shall come down with them, And the bullocks with the bulls; And their land shall be drunken with blood, And their dust made fat with fatness.”
“The wild oxen will come down with them, and the young bulls with the mighty bulls; and their land will be drunken with blood, and their dust made greasy with fat.”
Dragons that are jackals and sea creatures
Most of the King James "dragons" in the Old Testament translate tannim. That is the Hebrew word for jackals, the animals that haunt ruins. Jerusalem becomes "a den of dragons" in the King James and "a dwelling place of jackals" in the World English Bible. Where the Hebrew has tannin, the word means a serpent or great sea creature, as in Psalm 148.
“And I will make Jerusalem heaps, and a den of dragons; and I will make the cities of Judah desolate, without an inhabitant.”
““I will make Jerusalem heaps, a dwelling place of jackals. I will make the cities of Judah a desolation, without inhabitant.””
“Praise the Lord from the earth, ye dragons, and all deeps:”
“Praise Yahweh from the earth, you great sea creatures, and all depths;”
Gives re'em as the wild ox; tan (plural tannim) as the jackal; tannin as a serpent, dragon or sea monster; and tsepha / tsiph'oni as a kind of poisonous snake.
The cockatrice is a viper
A cockatrice is a creature of legend, a serpent said to hatch from a cock's egg and to kill with a look. In Isaiah the Hebrew has a word for a poisonous snake. Modern Bibles print "viper".
“And the sucking child shall play on the hole of the asp, and the weaned child shall put his hand on the cockatrice’ den.”
“The nursing child will play near a cobra’s hole, and the weaned child will put his hand on the viper’s den.”
Gives re'em as the wild ox; tan (plural tannim) as the jackal; tannin as a serpent, dragon or sea monster; and tsepha / tsiph'oni as a kind of poisonous snake.
Where the unicorn came from
The Septuagint, the Greek translation of the Jewish scriptures made before Christ, rendered re'em as "one-horned". Jerome, a Christian scholar of about AD 400, made the Latin Bible known as the Vulgate. It has "unicorns" in Isaiah and "dragons" for the desert animals of Isaiah and Jeremiah. English translators followed these older versions. The Hebrew itself has no such creatures.
The Septuagint, the Greek translation of the Hebrew Bible made before Christ, renders re'em at Numbers 23:22 with the Greek word for "one-horned" (monokeros), as it does in most of the verses where the King James Version has "unicorn". Brenton's English translation of the Greek accordingly reads "unicorn".
Jerome's Latin Bible, the Bible of the Western church for a thousand years, has unicornes ("unicorns") at Isaiah 34:7, dracones ("dragons") at Isaiah 34:13 and Jeremiah 9:11, and regulus, the basilisk of legend, at Isaiah 11:8. In other verses it renders re'em as rhinoceros.
What the NIV changed
The New International Version (NIV) puts wild oxen where the King James has unicorns, and vipers where it has cockatrices. It puts jackals, serpents or great sea creatures for most of the dragons. It keeps the dragon only where the text itself uses it as a symbol: in Revelation, where the dragon is the devil.
Where the King James Version has unicorns, the New International Version has wild oxen (for example Isaiah 34:7). Where the KJV has a cockatrice, the NIV has a viper (Isaiah 11:8, 14:29, 59:5). Most of the KJV's dragons become jackals, serpents or great sea creatures (for example Job 30:29, Psalm 91:13, Psalm 148:7). The dragon of Revelation stays a dragon.
“And he laid hold on the dragon, that old serpent, which is the Devil, and Satan, and bound him a thousand years,”
A book kept in its own words
The Qur'an says that God (﷾) Himself sent down the message and will guard it. He sent it down in Arabic (12:2), and that Arabic text is the Qur'an. A translation only explains its meaning and is never the Qur'an itself.
“We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption).”
“We have sent it down as an Arabic Qur'an, in order that ye may learn wisdom.”
Verse by verse
Numbers 23:22 — Torah, Bamidbar, chapter 23, verse 22
“God brought them out of Egypt; he hath as it were the strength of an unicorn.”
Cited for
The translators of 1611 were not putting myths into the Bible. "Unicorn" was their name for a real one-horned animal, perhaps the rhinoceros, and "dragon" could mean any great serpent.
The response
Whatever the translators pictured, the Hebrew names no unicorn, and most of its "dragons" are jackals (tannim). Re'em is a wild ox, as the Jewish translation of 1917 and the World English Bible both say. Where the Hebrew has tannin, a serpent or sea creature, "dragon" is near the mark. The word "unicorn" means one-horned, and it came from the old Greek and Latin Bibles, not the Hebrew. Later translators took these words out, because the Hebrew does not have them.
Where it lands — The Hebrew has wild oxen, not unicorns, and jackals where most of the "dragons" stand.
Perspectives
Mainstream Christian
Inspiration belongs to the Hebrew and Greek originals, not to any translation. The King James translators used the best knowledge of animals they had in 1611. Modern scholarship has improved the translation. The Hebrew never had unicorns, and most of its "dragons" are jackals.
The Hebrew word re'em means a wild ox, and modern translations render it correctly.
Rebuttal
Islam agrees: the Hebrew says wild ox, not unicorn. Yet the King James Version printed "unicorn" nine times. Islam goes further. It holds that the Qur'an itself is the Arabic God (﷾) sent down (12:2), so a translation only explains its meaning.
In 1611, "unicorn" and "dragon" were the translators' best names for real animals they did not know well. Examples are a one-horned wild beast and a great serpent.
Rebuttal
Where the Hebrew has tannin, a serpent or sea creature, "dragon" is near the mark. But most of the King James dragons are tannim, jackals, and re'em is a wild ox, not a one-horned beast. "Unicorn" came from the old Greek and Latin Bibles, not the Hebrew.
The Islamic perspective
The unicorns and cockatrices were never in the Hebrew, and most of the dragons were jackals. They came into the English Bible by way of Greek and Latin versions and stayed for centuries. The Hebrew names a wild ox and jackals. The Qur'an is recited in its own Arabic, and God (﷾) promised to guard it.
The King James Version has "unicorn" nine times where the Hebrew has re'em, the wild ox.
Most of the Old Testament "dragons" in the King James Version translate the Hebrew word for jackals.
The one-horned beast entered the Bible through the Greek Septuagint and the Latin Vulgate.
The NIV has removed almost all of these creatures. It keeps the dragon only as a symbol of the devil in Revelation.
The Qur'an says God (﷾) Himself guards the message He sent down.