I would like to ask your explanation regarding the "spirit" of man. After a man dies, the flesh goes to the ground, the soul goes to Heaven if the man has accepted Christ as Lord and personal Saviour, or to Hell if he has not, and the spirit, where does it go? In Ecclesiastes, it was mentioned there that the spirit goes to God who created it. Also the "spirit" of man is the one that gives the believer the knowledge of God through the Holy Spirit as Given in 1 Corinthians chapter 2. My first question is this, does the "spirit" of an unbelieving man have knowledge of God. Is it alive (the spirit) while man has not accepted Christ?

Because your questions are a bit involved, I am breaking your questions up into their different parts and answering one at a time.

So that I might not mislead, I need to explain a couple of things. I do not approach the doctrine of the spirit of man as many do. Certain scriptures have convinced me of a couple of twists that are not commonly taught and your questions hit on those areas. I simply affirm that God's word is the absolute authority and anywhere I have missed God's true teaching, take God's word. Another point I want to make is that the word "spirit" (as many Bible words) is used in several ways in the Bible. It can simply refer to the "spirit" of liveliness or animation that man shares with animals. In some cases, it clearly refers to the inner nature of man and in this usage includes the soul. However, what we are speaking of here is the spirit of man as distinct from and complimentary to the soul and the body.

In this regard, the spirit of man is the part of man that comes from God, relates directly to God, and returns to God. Ecclesiastes 12:7Then shall the dust return to the earth as it was: and the spirit shall return unto God who gave it.
See All... speaks of the time of death as the time when the "spirit shall return unto God who gave it." God gave the spirit and the spirit will return to God. Man and beast are distinguished in this way. "Who knoweth the spirit of man that goeth upward, and the spirit of the beast that goeth downward to the earth?" (Ecclesiastes 3:21Who knoweth the spirit of man that goeth upward, and the spirit of the beast that goeth downward to the earth?
See All...).

However, the human spirit is distinguished from the human soul in that it has no personal identity in and of itself. The soul of the man is still the man even if that soul departs the body. However, the spirit does not maintain that personal identity. It is simply God's communication link in the individual. "The Spirit itself beareth witness with our spirit, that we are the children of God" (Romans 8:16The Spirit itself beareth witness with our spirit, that we are the children of God:
See All...). "Now we have received, not the spirit of the world, but the spirit which is of God; that we might know the things that are freely given to us of God" (1 Corinthians 2:12Now we have received, not the spirit of the world, but the spirit which is of God; that we might know the things that are freely given to us of God.
See All...). God leads us to follow after Him through our spirit. "If we live in the Spirit, let us also walk in the Spirit" (Galatians 5:25If we live in the Spirit, let us also walk in the Spirit.
See All...).

Also, the human spirit has no life of its own. All life comes from God, but this is especially true of the spirit. It receives all of its life directly and immediately from the Spirit of God. When the Spirit of God is not in a man, that man's spirit is dead. Salvation is described as a quickening from the dead (Ephesians 2:1And you hath he quickened, who were dead in trespasses and sins;
See All..., 5). We are told by Jesus Christ that it is "the spirit that quickeneth; the flesh profiteth nothing" (John 6:63It is the spirit that quickeneth; the flesh profiteth nothing: the words that I speak unto you, they are spirit, and they are life.
See All...). In salvation, the "law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death" (Romans 8:2For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death.
See All...). A lost man has a dead, impersonal spirit that came from God and will return to God. Yet, without God, it is useless.

1 Corinthians 2:14But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned.
See All... states, "But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned." This passage (1 Corinthians 2:9-16 [9] But as it is written, Eye hath not seen, nor ear heard, neither have entered into the heart of man, the things which God hath prepared for them that love him. [10] But God hath revealed them unto us by his Spirit: for the Spirit searcheth all things, yea, the deep things of God. [11] For what man knoweth the things of a man, save the spirit of man which is in him? even so the things of God knoweth no man, but the Spirit of God. [12] Now we have received, not the spirit of the world, but the spirit which is of God; that we might know the things that are freely given to us of God. [13] Which things also we speak, not in the words which man's wisdom teacheth, but which the Holy Ghost teacheth; comparing spiritual things with spiritual. [14] But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned. [15] But he that is spiritual judgeth all things, yet he himself is judged of no man. [16] For who hath known the mind of the Lord, that he may instruct him? But we have the mind of Christ.
See All...) deals with how the Spirit of God illuminates the spirit of the believer in order to teach him the things of God. However, the natural man cannot receive the things of the Spirit. The natural man is the lost man; the man who is what he is by nature alone without the entrance of the Spirit. He cannot understand the things of the Spirit of God because these things are "spiritually discerned." That is, they are understood by the inner illumination of the Spirit. The spirit of the lost man is dead because it has no connection with the Spirit of God and it is therefore of no value in understanding the things of God. That is the answer to your first question.

Does the "spirit" of man go where the soul goes after the man dies? If the soul goes to heaven, of course the "spirit" goes to heaven with God also. But if the soul goes to hell, does the "spirit" also goes to hell? My example of this is the "rich man" in the book of Luke 16:19-31 [19] There was a certain rich man, which was clothed in purple and fine linen, and fared sumptuously every day: [20] And there was a certain beggar named Lazarus, which was laid at his gate, full of sores, [21] And desiring to be fed with the crumbs which fell from the rich man's table: moreover the dogs came and licked his sores. [22] And it came to pass, that the beggar died, and was carried by the angels into Abraham's bosom: the rich man also died, and was buried; [23] And in hell he lift up his eyes, being in torments, and seeth Abraham afar off, and Lazarus in his bosom. [24] And he cried and said, Father Abraham, have mercy on me, and send Lazarus, that he may dip the tip of his finger in water, and cool my tongue; for I am tormented in this flame. [25] But Abraham said, Son, remember that thou in thy lifetime receivedst thy good things, and likewise Lazarus evil things: but now he is comforted, and thou art tormented. [26] And beside all this, between us and you there is a great gulf fixed: so that they which would pass from hence to you cannot; neither can they pass to us, that would come from thence. [27] Then he said, I pray thee therefore, father, that thou wouldest send him to my father's house: [28] For I have five brethren; that he may testify unto them, lest they also come into this place of torment. [29] Abraham saith unto him, They have Moses and the prophets; let them hear them. [30] And he said, Nay, father Abraham: but if one went unto them from the dead, they will repent. [31] And he said unto him, If they hear not Moses and the prophets, neither will they be persuaded, though one rose from the dead.
See All... (Lazarus and the rich man). The rich man was able to identify father Abraham and Lazarus while he is in Hell and in torments. Does this mean that the "spirit" of man who gives knowledge about God is still with the rich man.

I believe the spirit of the lost man also returns to God; even while his soul goes to hell. The rich man did not need his spirit to recognize Abraham and Lazarus while in hell. He only needed God to allow him to see and know them. God, by His choice, allowed the rich man to see, recognize, and speak to Abraham and Lazarus. This does not require a human spirit.

Last question is that, if the "spirit" of unbelieving people who died will go to heaven with God as mentioned in Ecclesiastes while his soul has gone to hell. Will there be a lot of "spirit" of unbelieving people whose souls are being tormented in hell?

I understand this problem. But, as I said, the spirit without the Spirit is an impersonal substance. It is as if the Lord installed a phone line in each person's heart that could communicate with God. However, it cannot work until the owner of the house allows the representative to come and establish the connection. But there comes a time when the trial time is over and the Lord removes the line. He is able to do this because it belongs to Him anyway.

One final illustration may help. When Jesus died, we know that his human body was placed in the sepulchre (Matthew 27:59-60 [59] And when Joseph had taken the body, he wrapped it in a clean linen cloth, [60] And laid it in his own new tomb, which he had hewn out in the rock: and he rolled a great stone to the door of the sepulchre, and departed.
See All...). We are also told that the prophecy in Psalm 16:10For thou wilt not leave my soul in hell; neither wilt thou suffer thine Holy One to see corruption.
See All... - "thou wilt not leave my soul in hell" - applies to Jesus Christ (Acts 2:22-32 [22] Ye men of Israel, hear these words; Jesus of Nazareth, a man approved of God among you by miracles and wonders and signs, which God did by him in the midst of you, as ye yourselves also know: [23] Him, being delivered by the determinate counsel and foreknowledge of God, ye have taken, and by wicked hands have crucified and slain: [24] Whom God hath raised up, having loosed the pains of death: because it was not possible that he should be holden of it. [25] For David speaketh concerning him, I foresaw the Lord always before my face, for he is on my right hand, that I should not be moved: [26] Therefore did my heart rejoice, and my tongue was glad; moreover also my flesh shall rest in hope: [27] Because thou wilt not leave my soul in hell, neither wilt thou suffer thine Holy One to see corruption. [28] Thou hast made known to me the ways of life; thou shalt make me full of joy with thy countenance. [29] Men and brethren, let me freely speak unto you of the patriarch David, that he is both dead and buried, and his sepulchre is with us unto this day. [30] Therefore being a prophet, and knowing that God had sworn with an oath to him, that of the fruit of his loins, according to the flesh, he would raise up Christ to sit on his throne; [31] He seeing this before spake of the resurrection of Christ, that his soul was not left in hell, neither his flesh did see corruption. [32] This Jesus hath God raised up, whereof we all are witnesses.
See All...). This is discussed in other questions on our site, but I believe with most of the saints for ages that His soul went to hell. That is what the verse says. And, finally, on the cross Jesus said, "Father, into thy hands I commend my spirit: and having said thus, he gave up the ghost" (Luke 23:46And when Jesus had cried with a loud voice, he said, Father, into thy hands I commend my spirit: and having said thus, he gave up the ghost.
See All...). His "ghost" is His spirit (as in Holy Ghost). He turned His spirit over to the Father even though His soul went temporarily into hell to conquer that dread place for us. This gives us insight on what must happen to the lost man when he dies--with the exception of the purposeful commending of the spirit of Jesus to the Father.