There are two very different schools of thought with which to approach your questions. First there is the notion that a person can lose their salvation, and hence, lost the Holy Spirit's indwelling in their lives. Second is the idea of "Once Saved Always Saved" or the principle of Eternal Assurance. I will try to provide both viewpoints in attacking your questions.
1. If the Holy Spirit is still in someone, Does that mean there is still a chance to be saved?? (As long as the Holy Spirit is still in someone)
I am going to assume (for all of these questions) that you are speaking of someone who at one time professed faith in Christ Jesus and has since fallen away from that faith (lost their salvation). Your first question is a kind of mixture of both camps. Essentially, it appears as if to state someone can be unsaved yet still possess the Holy Spirit within them. Both schools of thought would exclude this possibility. Those who believe salvation can be forfeited because of willful sin the Holy Spirit would be essentially "kicked out" of their lives. For those who hold to Eternal Security if the Holy Spirit is still present then salvation is already achieved and can not be nullified.
So, since the question, as phrased, is an impossibility in both systems the answer must be in the negative.
2. What exactly Does the Holy Spirit do and how do we know we Have him?? (Does It have to do with a good consience?)
A very good question and one that every believer wrestles with today and in Biblical times. In the Old Testament David often lamented at the distance He felt God was from him, and even implored the Lord not take His Spirit from him. In New Testament times Paul would earnestly pray for his malady to be removed from him (his "thorn in the flesh") and was met with silence and even rebuke from the Lord.
Today the Holy Spirit fulfills within the believer the office Christ pointed towards. The Holy Spirit first convicts us (that is, convinces us) of our sin, our separation from God that sin causes, and leads us to repentance. Second, the Holy Spirit will enlighten our minds of the Truth of God's Word and open us to the Spiritual world that heretofore was hidden from the carnal mind. The Holy Spirit will also pray on our behalf when we are so grieved by our situation or state we are unable to form words adequately up to the task. He will also comfort us when our own strength and endurance for pain or grief or stamina are at a physical end.
3.If Someone Feels rejected by God and feels absolutely hopeless, Is there still a chance to be saved If they are hopeless and still repent?
That rejected feeling is often times the Holy Spirit acting on our hearts! Something may have gone amiss in our walk with God, be it our prayer lives, our obedience to His Word (Jesus), even sin that has crept back into our thoughts or deeds. If that is the case, we haven't lost salvation but we have lost communion with the Lord. That is, our closeness with Him, through the person and work of the Holy Spirit, is strained. Rather than seek to be resaved, a restoration of the relationship may be all that's needed to sooth a troubled spirit.
Hope this helps answer the questions a bit.