The Son of God, our Lord and Savior Jesus Christ, does not depend on the Father for his existence, as you suggest.
True enough, during his earthly sojourn, Jesus took great delight in doing the Father's will (see John 4:34; cf. John 17:4), even at great cost to himself, to the point, in fact, of shedding his lifeblood as the sacrificial Lamb of God who through his death took away the sin of the world (John 1:29 and 36).
In eternity past, however, Jesus, along with the Father and the Holy Spirit, was self-existent by virtue of his deity. God in fact has existed from all eternity as Father, Son, and Holy Spirit. The very notion of familial relationship originated in the Triune God by virtue of the Father-ship and Son-ship of the first and second persons of the Triune God.
Christians must not be too literal, as are Jehovah's Witnesses and Muslims, in the way they interpret such honorific titles of Jesus as
Jehovah's Witnesses will say, "See, Jesus is the 'firstborn,' so he must have had a beginning, and if he had a beginning he couldn't possibly be God." Muslims, on the other hand, dismiss the entire concept of divine Son-ship, thinking that the very notion that God has a Son is blasphemous. To them, associating progeny with God is unthinkable. God's greatness, they believe (ٱللهُ أَكْبَر Allāhu Akbar), obviates having children.
To Christians, however, the honorifics "Son of God," "firstborn of all creation," and "only begotten Son" do not denote--nor do they imply--that God had progeny with a beginning; rather, they indicate the elevated status of the Second Person of the trinity.
True to its roots in the Jewish Scriptures, the Tanakh, the belief in primogeniture meant that to the firstborn child (Hebrew בְּכוֹר bəḵōr) went the father's (and sometimes the mother's) inheritance. If God blessed the father with more children, the firstborn would still get a "double portion," indicating the favored status of the firstborn son.
All this to say: the Son-ship of Christ is eternal, with no beginning and no end. All of creation owes its existence to him (see John 1:3), and one day the cosmic and eternal hierarchy will exist forever. As Paul phrased it,
20 But now Christ has been raised from the dead, the first fruits** of those who are asleep. 21 For since by a man came death, by a man also came the resurrection of the dead. 22 For as in Adam all die, so also in [h]Christ all will be made alive. 23 But each in his own order: Christ the first fruits**, after that those who are Christ’s at His coming, 24 then comes the end, when He hands over the kingdom to the God and Father, when He has abolished all rule and all authority and power. 25 For He must reign until He has put all His enemies under His feet. 26 The last enemy that will be abolished is death. 27 For He has put all things in subjection under His feet. But when He says, “All things are put in subjection,” it is evident that He is excepted who put all things in subjection to Him. 28 When all things are subjected to Him, then the Son Himself also will be subjected to the One who subjected all things to Him, so that God may be all in all (1 Corinthians 15:20-28 NASB, my emphasis).
And, more simply and economically,
Christ is the head of every man, and the man is the head of a woman, and God is the head of Christ (1 Corinthians 11:3 NASB).
In conclusion, the clear teaching of Scripture is that Son-ship is an eternal relationship within the Godhead. Our human notion of son-ship is but a shadow of that which existed forever in the bosom of God the Father (see John 1:18). As to the Father-ship of God to those within the human race who acknowledge Christ for who he is and receive him into their bosom, that too will be an eternal relationship, and we have His word on that:
He [i.e., Jesus] came to His own [people, the Jews], and those who were His own did not receive Him. But as many as received Him, to them He gave the right to become children of God, even to those who believe in His name, who were born, not of blood nor of the will of the flesh nor of the will of man, but of God (John 1:11-13 NASB).