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 12:7 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 3:21).
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 8:16). "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 2:12). God leads us to follow after Him through our spirit. "If we live in the Spirit, let us also walk in the Spirit" (Galatians 5:25).
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 2:1, 5). We are told by Jesus Christ that it is "the spirit that quickeneth; the flesh profiteth nothing" (John 6:63). In salvation, the "law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death" (Romans 8:2). 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 2:14 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) 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 (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). We are also told that the prophecy in Psalm 16:10 - "thou wilt not leave my soul in hell" - applies to Jesus Christ (Acts 2:22-32). 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 23:46). 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.