Temporal terms used to describe eternal realities
When Christians speak of the Son being 'begotten' what they mean to convey is a relationship within the divine nature, between two aspects thereof, not a temporal event of literal begetting (as in the Qur'an, where God actually copulates with a divine consort to produce a son), or of something added to God which before He didn't have.
One must bear these limitations in mind.
Can one describe eternal relationships without using a temporal analogy, after all? I think not. All relationships on earth are described by way of a course of events which led to two things rather than one, etc. In God, these words of "beget" etc. are used only insofar as "beget" denotes full sharing of the nature of the begetter in that which is begotten. That's why we use it of the Son, because 'in Him dwells the fullness of the divinity' (Colossians 2:9). Could God ever be without thought, or ever without wisdom? This is how integral to the nature of God we believe the Son and Holy Spirit to be to the Father, the αρχη or source of this "Son" and his "Spirit"—'source' being used with the above qualification.
It's comparable to fire. One can address simply "fire" (just as we Chritsians address 'the one God'), yet one can distinguish between the overall fire ('God'), and the light and the heat, for example, emitted. You don't have fire without the light ('the Son') or the heat ('the Spirit), but the fire as distinct from light and from heat can nonetheless be distinguished ('the Father'). But even if we say the light is 'emitted from' the fire and 'heat created' we don't mean to imply that there could have ever been fire, to which was then added light, or added heat, without said aspects, but speak of the causal relationship only. So it is with the Trinity: the begetting of the Son only tells us that God's nature is fully in what we name Son, not that an event took place literally in which God was without a Son and then gave birth to a Son. Son denotes the presence of the nature of God in what is called Son, not a literal offspring.
In keeping with the analogy with fire, we could say that the Spirit is the heat generated by the fire, via the light it emits ('the Son' John 16:15), so that the light ('the Son') is united to the αρχη or source of fire and the fire to the light via this intermediary effect of 'both together' and yet the fire or 'the Father' is the source, ultimately of both—again, 'source' in the 'what is the relationship between these?' not 'was without them and then created them' (because language is used like beget which in humans involves time, but with God, because it denotes only the relationship, does not). Nothing in fire 'comes first' inasmuch as fire isn't present without light or heat. Noting in the Trinity comes first because, once again, relationships are described using terms like beget or proceed only insofar as they are useful for describing how one person relates to the other.
As to why these 'light' and 'heat' 'effects' of God are personal, it's simply that what is God is all-knowing, all-powerful, all-loving, etc. which are inherently personal things, and because whatever is eternal is God, and is synonymous with the nature of God by definition. There are no 'inanimate things' in God's nature.
To your question more specifically...
"God is not a man" (Numbers 23:19). If He were already a man, the incarnation (Latin for becoming flesh) wouldn't make sense. John 1:1 would read instead: "In the beginning was the Word, and the Word with God, and the Word was flesh: ... and the Word remained flesh, and made His dwelling among us."
Nor is Jesus a human puppet. Jesus means "Saviour" and was the name given to the boy born in Bethlehem of the Virgin Mary. He wasn't a created person, but an eternal divine Person, the Word, "whose origins are of old, even from the days of eternity" (Micah 5:2), who became a man in the sense that He took on a human nature, not that the divine nature was mutated into a human nature, or that He gave up being divine. Inasmuch as the Word took on a human nature and a human soul, He would faint, get hungry, and die: that was quite the purpose of the Redemption, to become a man, "suffer many things ...and be killed" (Mark 8:31): "for scarcely will a man die even for a just man: but God shows the love He has for us in that while we were still sinners, Christ died for us" (Romans 5:7-8).
There is a heresy called Docetism that says Jesus 'only seemed to be' human. This is rejected by historic Christianity as denying that Jesus has come in the flesh (2 John 1:7). A related heresy, and one you seem to have mistaken for the orthodox doctrine is Apollinarianism, which says Jesus didn't have a human soul, but was 'possessed' by the Word (like a puppet), instead of the Word having a human soul created to live like a normal human being, who vivifies and lives in His body by a human soul, while personally being the Word made a man.
What is the Spirit?
The Spirit is the mutual recognition of the Father and Son, which is often simply called the Love between the Father and the Son. They know each other to be the perfect God, sharing and being one nature, and so can only love each other; this is the Holy Spirit, the 'Sigh' ('רוח') of love between the two.
See the point about about Him being personal, and about none of this taking place in time, but being an eternal reality described using relational terms.