God's Spirit is not an expression found in the Bible.
Spirit is God very nature. The quiddity of a Thing. The Infinite Divine Essence itself, for "God is Great without Quantity and Good without Quality". Therefore when Holy and Spirit is together it signifies The Hypostasis (Subsisting/ Substantive Reality),of The Holy Spirit in distinction and relation to The Father and The Son .So not rocket science.
God having a spirit even tho He is Spirit by His very Nature does not Negate a distinction of Personalis of whom it is spoken in The Divine. It is quite clear God The Father extends Hypostasis to another , To The Son, and by The Son to The Holy Spirit.
Unless I am quoting ,I never feel like typing the 's . I assume many know already, so sue me.
NO I cannot.
God IS A Spirit; He does NOT have a Spirit.
Number said of God is only in the human mode of intelligibility as when we say, One of the Father, another of The Son, and another of The Holy Spirit ,but not as if there exist any real supposition or subjection( ie composite subjects/beings/gods. We number/place number in The Unity of Nature, so as not to confound and confuse The Subsistence- ie Threefold Manner of Existence, like Oneness insist on doing.
And who the Son chooses to reveal Him to. Try to read the scripture from the context and not from your preconceived idea.
Which means The Holy Spirit,
The Spirit of God, The Spirit of The Father, The Spirit of The Lord, The Spirit of Christ, etc. etc.