It’s late and I’m kind of worn down, but I’ll give this a shot.
One explanation of the Holy Spirit that I like is that the Spirit flows out from the divine love and unity shared by the Father and the Son. As Christians, the presence of the Spirit in our lives enables us to partially partake in that love, for Christ is in the Father and we are in Christ (John 14:20). St. Paul supports this view when he refers to the Spirit as the first part of our “inheritance” (Eph. 1:13-14). Our eventual goal is to be in complete union with God in heaven; the gifts of the Spirit are the first step towards achieving that union. By allowing us to partake in His divine life while we’re on this earth, God strengthens us with His nature and gifts, acting as an Advocate who brings us closer to Him (1 John 2:1, etc.).
So, applying this to your questions:
Can we agree that we are comprised of a soul and a spirit? The spirit is the life in man and the soul being will or consciousness. Then, what is the difference between this spirit in man and the Holy Spirit? Do we always have the Holy Spirit and does it exist in all things?
Our souls are (obviously!) distinct from God’s divine love. But God chooses to infuse us with His nature through the gift of the Holy Spirit. The entities are distinct, but God’s Spirit works to strengthen our own souls.
As for where the Spirit exists: I’m just speculating, but I would say that the Spirit is omnipresent insofar as that God is omnipresent and the Spirit is an integral part of God. On the other hand, the infusion of the Spirit’s gifts into our own lives is not automatic: We have to have a relationship with God, initiated through Baptism, before we can enter into His divine life.
Does the Holy Spirit exist in us, even before we were baptized?
I would venture (and again, this is speculation) that although God leads us to Baptism by His grace, the gifts of the Spirit (which, again, are a partial union with God) aren’t available to us until we’ve received the sacrament and are Christians. (If the Spirit is the first part of “our inheritance,” as Paul teaches in Ephesians 1, then its gifts are at least primarily given to Christians, since only Christians have access to the inheritance Paul is talking about.)
How does this correspond with the nature of Baptism?
Since Baptism initiates us into the Christian life and infuses us with God’s grace, it enables us to receive God’s gifts, including the gifts of the Spirit. These gifts and our ability to partake in them are then strengthened by Confirmation.
Hope that helps you in some way (and that I didn’t accidentially trip up and say something distinctly heterodox!).