This is to address a part of OP's question which does not yet seem addressed in the existing answers. He asks about differences between the KJV and traditional Catholic Bibles, and whether the KJV was "fundamentally changed" to fit a Protestant perspective.
The Douay-Rheims Version, as Ken Graham points out, was completed before the KJV and this was the traditional Catholic Bible against which the KJV can be compared. There were differences and it was the view of English-speaking Catholics that they were the result of errors in the KJV, and that these errors fitted a Protestant perspective, and that some, at least, were deliberate. Protestants usually took the opposite view, claiming it was the Douay-Rheims version (DRV) which had deliberate errors in it.
Thomas Ward, a seventeenth century English convert to Catholicism, published a book in 1688 called "Errata of the Protestant Bible" listing numerous differences and explaining their significance. This edition, with some additional material, was published in Dublin in 1841. It deals with "errors" in the KJV but also many in earlier versions which were corrected in the KJV (the last column in his tables shows the KJV position). It is written in the highly polemical style typical of its time.
Below are just some of these, with a summary of their significance (my attempted summary of what Ward wrote at much greater length) In each case I will put the Douay-Rheims verse first, followed by the King James. Many are very subtle.
Genesis 14 v18
But Melchisedech the king of Salem, bringing forth bread and wine, for he was the priest of the most high God,
And Melchizedek king of Salem brought forth bread and wine: and he was the priest of the most high God.
Here the DRV shows bread and wine as intrinsic to Melchisedech's priesthood, whereas the KJV suggests the fact Melchisedech was a priest, and the fact he brought forth bread and wine are two possibly unrelated facts.
1 Corinthians 11 v27
Therefore whosoever shall eat this bread, or drink the chalice of the Lord unworthily, shall be guilty of the body and of the blood of the Lord
Wherefore whosoever shall eat this bread, and drink this cup of the Lord, unworthily, shall be guilty of the body and blood of the Lord
The difference here is that Protestants received both the bread and wine whereas Catholics generally received only bread (communion in one kind only). The different translations of this verse seem to reflect this difference in practice.
Jeremiah 11 19
And I was as a meek lamb, that is carried to be a victim: and I knew not that they had devised counsels against me, saying: Let us put wood on his bread, and cut him off from the land of the living, and let his name be remembered no more.
But I was like a lamb or an ox that is brought to the slaughter; and I knew not that they had devised devices against me, saying, Let us destroy the tree with the fruit thereof, and let us cut him off from the land of the living, that his name may be no more remembered
This verse was interpreted by Jerome as a Eucharistic prophecy of the Crucifixion in which Christ's body is referred to as "bread". The Protestant translation by referring to "fruit" rather than "bread" negates this interpretation.
Luke 3 v3
preaching the baptism of penance for the remission of sins
preaching the baptism of repentance for the remission of sins
Acts 2 v38
Peter said to them: Do penance, and be baptized every one of you in the name of Jesus Christ,
Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ
In each of these verses, one referring to the message of John the Baptist and the other to the events of the first Whitsun, the Catholic Bible has penance while the Protestant one has only repentance.
1 Corinthians 9 v5
Have we not power to carry about a woman, a sister, as well as the rest of the apostles
Have we not power to lead about a sister, a wife, as well as other apostles
Here St Paul is saying that as an apostle he has the right to a lady companion, but only in the Protestant Bible is it stated she could be his wife.
Luke 1 v28
And the angel being come in, said unto her: Hail, full of grace, the Lord is with thee: blessed art thou among women
And the angel came in unto her, and said, Hail, thou that art highly favoured, the Lord is with thee: blessed art thou among women
There is at least a difference in degree between being full of grace and being highly favoured.
Worship of Inanimate Objects
Psalm 99 v5 (Psalm 98 according to the Catholic reckoning)
Exalt ye the Lord our God, and adore his footstool, for it is holy
Exalt ye the LORD our God, and worship at his footstool; for he is holy.
Hebrews 11 21
By faith Jacob dying, blessed each of the sons of Joseph, and adored the top of his rod
By faith Jacob, when he was a dying, blessed both the sons of Joseph; and worshipped, leaning upon the top of his staff
The DRV refers to adoration of the footstool and the rod, whereas the KJV refers to the footstool and rod as locations, not objects, of worship.
Limbus Patrum (the Limbo of the Fathers)
Genesis 42 v38
you will bring down my gray hairs with sorrow to hell
then shall ye bring down my gray hairs with sorrow to the grave.
Here, Jacob is fretting about what will happen to him if he allows Benjamin to travel to Egypt and he does not come back. According to the DRV he expects, on death, his soul will go to hell, whereas according to the KJV he is talking about the burial of his body. There are several similar verses.
Malachi 2 v7
For the lips of the priest shall keep knowledge, and they shall seek the law at his mouth: because he is the angel of the Lord of hosts.
For the priest's lips should keep knowledge, and they should seek the law at his mouth: for he is the messenger of the LORD of hosts
An interpretation of this verse is that it is a prophecy of the Papacy. The DRV says what "shall" happen, a definite statement about the future, the KJV says what "should" happen but with no guarantee that it will.
1 Peter 2 13
Be ye subject therefore to every human creature for God's sake: whether it be to the king as excelling; Or to governors ...
Submit yourselves to every ordinance of man for the Lord's sake: whether it be to the king, as supreme; Or unto governors ...
Here the DRV describes the king as "excelling", the KJV asserts that he is "supreme". Not all Protestants believe in Royal Supremacy, but the Church of England does, and King James certainly did (as indeed did several Catholic monarchs). The Catholic Church regarded the Pope as supreme.
1 Timothy 4 14
Neglect not the grace that is in thee, which was given thee by prophesy, with imposition of the hands of the priesthood.
Neglect not the gift that is in thee, which was given thee by prophecy, with the laying on of the hands of the presbytery
Acts 14 v23
And when they had ordained to them priests in every church, and had prayed with fasting, they commended them to the Lord, in whom they believed.
And when they had ordained them elders in every church, and had prayed with fasting, they commended them to the Lord, on whom they believed.
The DRV refers to Christian priests being ordained by Paul and Barnabas whereas the KJV calls them elders. The DRV refers specifically to ordination conferring grace, which the KJV terms a gift, which may reflect a more sacramental understanding of ordination in the Catholic Church. There are several other verses where the DRV says "priest" and the KJV has "elder". The KJV also has "overseer" in some of the places where the DRV has "bishop", although the KJV does refer to bishops (e.g. 1 Timothy 3 v1).
Salvation by Faith
Luke 18 42
And Jesus said to him: Receive thy sight: thy faith hath made thee whole
And Jesus said unto him, Receive thy sight: thy faith hath saved thee
The DRV has Jesus talking of physical healing whereas the KJV suggests salvation, as a result of faith.
The King James Version differed in numerous, often very subtle, ways from the DRV on points of disputed doctrine.
The circumstances of its translation, and the description of the Pope as "that man of sin" in the Epistle Dedicatory (translators letter to James) hardly endeared it to the Catholic Church. Also there was already a Catholic translation in the DRV so the KJV would, in any case, have been superfluous. But beyond all these reasons there were genuine doctrinal differences between the two.