1. "Christianity neither is, nor ever was a part of the common law"
Jefferson was referring to the common law of England which he identified as the collection of laws which existed prior to the Magna Carta. He was not referring to the laws of America. Here is the context from which your quote was taken:
"For we know that the common law is that system of law which was introduced by the Saxons on their settlement in England, and altered from time to time by proper legislative authority from that time to the date of Magna Charta, which terminates the period of the common law, or lex non scripta, and commences that of the statute law, or Lex Scripta. This settlement took place about the middle of the fifth century. But Christianity was not introduced till the seventh century; the conversion of the first christian king of the Heptarchy having taken place about the year 598, and that of the last about 686. Here, then, was a space of two hundred years, during which the common law was in existence, and Christianity no part of it. If it ever was adopted, therefore, into the common law, it must have been between the introduction of Christianity and the date of the Magna Charta. But of the laws of this period we have a tolerable collection by Lambard and Wilkins, probably not perfect, but neither very defective; and if any one chooses to build a doctrine on any law of that period, supposed to have been lost, it is incumbent on him to prove it to have existed, and what were its contents. These were so far alterations of the common law, and became themselves a part of it. But none of these adopt Christianity as a part of the common law. If, therefore, from the settlement of the Saxons to the introduction of Christianity among them, that system of religion could not be a part of the common law, because they were not yet Christians, and if, having their laws from that period to the close of the common law, we are all able to find among them no such act of adoption, we may safely affirm (though contradicted by all the judges and writers on earth) that Christianity neither is, nor ever was a part of the common law."
2. "Christianity is the most perverted system that ever shone on man"
Jefferson never said this. What he actually said was:
"This was the real ground of all the attacks on you. Those who live by mystery & charlatanerie, fearing you would render them useless by simplifying the Christian philosophy,– the most sublime & benevolent, but most perverted system that ever shone on man,– endeavored to crush your well-earnt & well-deserved fame."
This is an excerpt from a letter that Jefferson wrote to a dissenting minister from England named Joseph Priestley. Both Jefferson and Priestley believed that the doctrines preached by Jesus Christ had been corrupted by the mainline churches of Christianity. Priestly even published a book entitled "An History of the Corruptions of Christianity" in which he attempted to trace how various perversions of Christ's original doctrines had taken place throughout history. Neither Jefferson nor Priestly claimed that Christianity was corrupt in itself. Rather, they argued that the mainline churches had perverted the gospel of Christ just as Paul had warned in Galatians 1:7. Jefferson viewed Christianity as a sublime and benevolent religion which had often been perverted by those desiring to twist it to fit their own purposes.
3. "The government of the United States is not in any sense founded on the Christian Religion."
This was never said by Adams or any other American. It comes from a very poor attempt to translate from Arabic into English a letter written by the Dey of Agiers to the Pasha of Tripoli. I devoted an entire chapter of my book Hidden Facts of the Founding Era to discussing this quote and the many errors surrounding it. You can read that chapter online for free at this link.
4. "All national institutions of churches, whether Jewish, Christian, or Turkish, appear to me no other than human inventions set up to terrify and enslave mankind, and monopolize power and profit."
Thomas Paine had practically no influence on the formation of America. He was hated and despised by most Americans, and when the founding fathers discovered that he had been imprisoned in Parris during the French Revolution, they wrote to each other that it would be best if he spent the rest of his years rotting in a French cell. Paine's only claim to influence in the American Revolution was the pamphlet "Common Sense," but that pamphlet was actually commissioned by Dr. Benjamin Rush who oversaw it's writing and recommended it to other leaders of the Revolution once it met with his approval.
5. "Religion and government will both exist in greater purity the less they are mixed together."
Madison was discussing the question of whether we ought to have an established religion. Here is the context from which your quote was taken:
"Notwithstanding the general progress made within the two last centuries in favour of this branch of liberty, & the full establishment of it, in some parts of our Country, there remains in others a strong bias towards the old error, that without some sort of alliance or coalition between Govt. & Religion neither can be duly supported.
"Such indeed is the tendency to such a coalition, and such its corrupting influence on both the parties, that the danger cannot be too carefully guarded agst. And in a Govt. of opinion, like ours, the only effectual guard must be found in the soundness and stability of the general opinion on the subject.
"Every new & successful example therefore of a perfect separation between ecclesiastical and civil matters, is of importance. And I have no doubt that every new example, will succeed, as every past one has done, in shewing that religion & Govt. will both exist in greater purity, the less they are mixed together.
"It was the belief of all sects at one time that the establishment of Religion by law, was right & necessary; that the true religion ought to be established in exclusion of every other; And that the only question to be decided was which was the true religion ... The example of the Colonies, now States, which rejected religious establishments altogether, proved that all Sects might be safely & advantageously put on a footing of equal & entire freedom; and a continuance of their example since the declaration of Independence, has shewn that its success in Colonies was not to be ascribed to their connection with the parent Country. If a further confirmation of the truth could be wanted, it is to be found in the examples furnished by the States, which have abolished their religious establishments."
Madison was not objecting to the idea of establishing a government on religious principles which is what most Christian mean when they claim that America was founded as a Christian nation. He was simply opposed to the establishment of a state religion. By the way, the concept of a separation between church and state is itself a religious concept which was introduced into America by the Baptists. As I wrote in my book "Hidden Facts of the Founding Era":
The concept of true freedom of religion was introduced in America by the Baptists of Rhode Island. George Bancroft, the famous American historian, recorded that in November of 1658, the colonists of Rhode Island requested that Roger Williams plead their case before the king that they would 'not be compelled to exercise any civil power over men's consciences.' They declared that it was their goal 'to hold forth a lively experiment, that a most flourishing civil state may stand, and best be maintained, with a full liberty of religious concernments.' The king consented to their request, and in July of 1663, he signed the Charter of Rhode Island which included the following statement:
'Our royal will and pleasure is, that no person within the said colony, at any time hereafter, shall be any wise molested, punished, disquieted, or called in question, for any differences in opinion in matters of religion.'
Then, in May of 1664, the legislature of that colony passed the first law in American history which established true religious freedom. That law stated that:
'No person shall at any time hereafter be any ways called in question for any difference of opinion in matters of religion.'
In May of 1665, the legislature reaffirmed this law with a statement declaring that religious freedom had been granted in Rhode Island ever since Roger Williams began the settlement there in 1636. Their statement said:
'Liberty to all persons, as to the worship of God, had been a principle maintained in the colony from the very beginning thereof; and it was much in their hearts to preserve the same liberty forever.'
And then in 1680, they issued yet another statement declaring:
'We leave every man to walk as God persuades his heart; all our people enjoy freedom of conscience.'
...After relaying the above facts about Rhode Island, Mr. Bancroft wrote: 'Freedom of conscience, unlimited freedom of mind, was, from the first, the trophy of the Baptists.'
By the way, Roger Williams and the Baptists of Rhode Island were influenced by Thomas Helwys, a Baptist preacher in England who wrote the earliest recorded defense of religious freedom in the English language.