An explanation from Calvinist about why Cain's offering is not accepted by God?

Why does God approve of Abel as righteous because of his offering?

You stated this in point 5 of your question, and though it wasn’t explicitly asked, I wanted to address it because it is contrary to our first principles. Conversely, God approves of Abel’s offering because of his righteousness- a righteousness, like all godly righteousness, that comes to him by faith. When Calvin says, "God begins with the person of the offerer" he means that the value of the offering is that it is an expression of adoration, and not that it is an attempt to gain approval; the value of the sacrifice exists solely in the heart of the offerer, because the approval of God cannot be earned, except so perfectly.

What is the cause of Cain's offering not being accepted by God ?

In Book 2 of Calvin’s Institutes of the Christian Religion, Calvin discusses the episode of Cain and Abel. He offers an interpretation of Genesis 4:7 that is not common, even among modern translations of the passage, acknowledging the error of Catholic translators.

They adduce the passage of Genesis, “Unto thee shall be his desire, and thou shalt rule over him,” (Gen. 4:7). This they interpret of sin, as if the Lord were promising Cain that the dominion of sin should not prevail over his mind, if he would labour in subduing it. We, however, maintain that it is much more agreeable to the context to understand the words as referring to Abel, it being there the purpose of God to point out the injustice of the envy which Cain had conceived against his brother. And this He does in two ways, by showing, first, that it was vain to think he could, by means of wickedness, surpass his brother in the favour of God, by whom nothing is esteemed but righteousness; and, secondly, how ungrateful he was for the kindness he had already received, in not being able to bear with a brother who had been subjected to his authority.

Now, it helps to remember that the Reformers, among other complaints, rejected the teaching of the Roman Church that justification is a consequence of godly living. The concern of the Catholic presbyters was the moral living of their congregants and their dedication to the authority of Rome, and their teaching reflected that concern. It is not surprising that a person with such a perspective regarding righteousness would naturally adopt an interpretation that God spoke to Cain, saying that he ought to overcome his temptation through labour. To the contrary, Calvin posits that God was prophesying the very domination over Abel that Cain was later to display, in a similar way that He demolishes the pride of the wicked throughout the narrative that follows Genesis, which culminates in his demolition of all wickedness.

Calvin continues, giving the objector the benefit of the doubt by supposing that 4:7 does refer to sin. He posits then that the words constitute either an order or a promise, and examines both cases. In so doing, he addresses your second question.

Has God regenerated Cain and gave him faith by the time God speak to Cain as read in verse 6 and 7 ?

If God had regenerated Cain, he would have been able to speak the words in that passage as a promise, that is, offering assurance. It is common throughout the scriptures for God, through visions, prophets, or apostles, to offer the assurance of justification- a promise- to the elect. Calvin considers this as he continues.

If so, his words contain either an order or a promise. If an order, we have already demonstrated that this is no proof of man’s ability; if a promise, where is the fulfillment of the promise when Cain yielded to the sin over which he ought to have prevailed?

Because Cain fails, Calvin concludes that the passage cannot be a promise, lest the word of God fail, and must therefore be a command, then makes a statement that I believe answers your question.

For, if the dominion spoken of refers to sin, no man can have any doubt that the form of expression is imperative, declaring not what we are able, but what it is our duty to do, even if beyond our ability. Although both the nature of the case, and the rule of grammatical construction, require that it be regarded as a comparison between Cain and Abel, we think the only preference given to the younger brother was, that the elder made himself inferior by his own wickedness.

But surely Cain had faith. One might ask, how could a man such as Cain, who heard the voice of God, lack faith? What is the difference between the faith of Cain and that of Abel?

In the third book of Institutes, On Faith, Calvin answers.

But since the heart of man is not brought to faith by every word of God, we must still consider what it is that faith properly has respect to in the word. The declaration of God to Adam was, “Thou shalt surely die,” (Gen. 2:17); and to Cain, “The voice of thy brother’s blood crieth unto me from the ground,” (Gen. 4:10); but these, so far from being fitted to establish faith, tend only to shake it. At the same time, we deny not that it is the office of faith to assent to the truth of God whenever, whatever, and in whatever way he speaks: we are only inquiring what faith can find in the word of God to lean and rest upon. When conscience sees only wrath and indignation, how can it but tremble and be afraid? And how can it avoid shunning the God whom it thus dreads? But faith ought to seek God, not shun him. It is evident, therefore, that we have not yet obtained a full definition of faith, it being impossible to give the name to every kind of knowledge of the divine will.

It is not the practice of faithfulness or measure of faith that is concomitant with the approval of God, but indeed a faith with a specific object and nature. God did not approve of Cain’s sacrifice because with it he sought the favor of God, and even to surpass the favor of his brother. Abel on the other hand loved God and did not seek righteousness with his sacrifice, but offered it as an expression of his adoration. Cain’s heart was hard, and so though he had “faith” in the sense of knowledge of the existence and identity of God, he lacked faith in the sense of knowledge that God himself is the sole source of righteousness by favor, that is, by grace, and that his favor is not obtained as recompense by sacrifice and labor, but a gift given according to trust and love that are the product of illumination by his Spirit.

Calvin returns to the episode in his commentary on Hebrews 11:4, saying,

The Apostle’s object in this chapter is to show, that however excellent were the works of the saints, it was from faith they derived their value, their worthiness, and all their excellences; and hence follows what he has already intimated, that the fathers pleased God by faith alone.

Now he commends faith here on two accounts, — it renders obedience to God, for it attempts and undertakes nothing, but what is according to the rule of God’s word, — and it relies on God’s promises, and thus it gains the value and worth which belongs to works from his grace alone. Hence, wherever the word faith is found in this chapter, we must bear in mind, that the Apostle speaks of it, in order that the Jews might regard no other rule than God’s word, and might also depend alone on his promises.

He says, first, that Abel’s sacrifice was for no other reason preferable to that of his brother, except that it was sanctified by faith: for surely the fat of brute animals did not smell so sweetly, that it could, by its odor, pacify God. The Scripture indeed shows plainly, why God accepted his sacrifice, for Moses’s words are these, “God had respect to Abel, and to his gifts.” It is hence obvious to conclude, that his sacrifice was accepted, because he himself was graciously accepted. But how did he obtain this favor, except that his heart was purified by faith.