Why didnt Cain respond like Abel? Why didnt God accept his offering? What did God mean when He said, sin lieth at the door? And what is the way of Cain spoken of in the New Testament?

Based upon what is set forth later on in Hebrews 11 regarding Cain and Abel we can determine that faith is the issue, just as Hebrews 11:4 states. Romans 10:17, along with Hebrews 11, make clear, faith cometh by hearing, and hearing by the word of God. Abel responded positively to what he had heard God say. The same verse by its wording makes it evident that Cain also heard what God said. (In addition, based upon the rhetorical nature of the question God asked Cain at the beginning of Gen.4:7, it is evident that God  had already made known to Cain what he was to do when he knew he didnt do well, and that Cain himself knew this to be the case.) So on the basis of this we can be persuaded that God had told both Abel and Cain that if they doest well they would be accepted, but if not, then the issue was for them to bring the animal sacrifice. They both knew what it meant to doest well in Gods sight. They both knew how God wanted them to respond if they did not doest well. Abel, knowing that he did not doest well by nature, believed what God said and brought the sacrifice God had specified. Cain, however, did not believe what God had said. Not only did Cain not believe that he was guilty in Gods sight, but he actually thought himself to be suitably righteous. Accordingly, he did not bring the animal sacrifice, but he brought an offering of the work of his own hands as a tiller of the ground. As Hebrews 11:6 says, but without faith it is impossible to please him, Cain, therefore, clearly did not respond in faith at all to what God had said, and hence he did not please God at all. Cains blatant unbelief is why God had no respect unto his offering. Now specifically regarding what God said to Cain in Genesis 4:7, it is clear God reproved Cains negative response of unbelief. This He did by His three rhetorical questions set forth in Gen.4:6-7. Again, Cain knew what God had said about being accepted and about the sacrifice to be brought and offered if he did not doest well.When God did not have respect unto Cains offering he became very wroth, and his countenance fell, his wrath and fallen countenance were the result of his wounded pride, for he thought himself righteous. When God said unto Cain, why art thou wroth? And why is thy countenance fallen? If thou doest well, shalt thou not be accepted? (Gen.4:6-7), He that searcheth the hearts and triest the reigns, and declareth unto man what is his thought, and that leadeth to repentance, was making known unto Cain the thoughts that were in him and was leading him to repentance. What Cain needed to do was repent of his erroneous thinking. In connection with this God goes on to say unto him in the rest of verse 7, and if thou doest not well, sin lieth at the door. And unto thee shall be his desire, and thou shalt rule over him. When God says to Cain that sin lieth at the door, He is talking about the fact that the sacrifice for sin that Cain could offer lieth at the door right there in front of Cain, so to speak. (This door was specifically the very door of the Garden of Eden where the Cherubims were, along with the flaming sword that kept the way of the tree of life. This same basic set up also existed later on with the tabernacle, at the door of which the animals were brought. There the animal was identified as the bringers substitute, receiving the bringers sinful identification, and as such being made sin for him. It all took place at the door.) The same situation existed at the door of the garden. With God leading Cain unto repentance and in saying to him that sin lieth at the door, He was telling him that the animal would be the willing substitute-bearer of his sin, and made sin for him, was lying right there at the door where he was standing. The animal truly was willing. Hence, God said to Cain, and unto thee shall be his desire, and thou shalt rule over him. Cain, however, disdainfully hardened his heart to Gods words and refused to change his mind. Yet Cain also did more than this. He also persued a hate laden course of action that was both generated and fueled by his highly offended pride. In hate he lashed out at Gods counsel and mercy, bringing it to bear upon Abel as he murdered him. And with Cain, the Adversary established the first of three particular negative responses to the word of God, which when progressively followed result in the followers becoming Satan-like. As such Cain became the founder, so to speak, of the way of Cain, which was later on followed by Israels vain religious leadership after they hardened their hearts to the councel of God against themselves contained in the gospel of the kingdom. They too thought themselves righteous, rejected the councel of God against themselves, and despised the mercy of God shown to them. In so doing, they persued the same hate laden course of action generated and fueled by their highly offended pride. In hate they lashed out at the righteous remnant of Israel. And this way is still followed in this dispensation, when religious leaders deny that sin lieth at the door and try to establish a religion based upon their own desire of justice. They rise up in hatred against their fellow man and murder him when they try to pretend that Gods sacrifice of His Son was not good enough to erase all sin debt. They disbelieve that Jesus is the Savior of all men.