According to the doctrine of Trinity, was Jesus always co-equal to the Father?

The Trinity is expressed best in 1 John 5:7. For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.

Becoming the man Jesus He did not give up any of his attributes. Philippians 2:6-8  Who, being in the form of God, thought it not robbery to be equal with God: But made himself of no reputation, and took upon him the form of a servant, and was made in the likeness of men: And being found in fashion as a man, he humbled himself, and became obedient unto death, even the death of the cross.

God's Word describes Jesus as: 1. existing "In the beginning"... "with God" 2. Being the creator verse 2 "All things were made by him" (John 1:10) 3. And as the source of life, able to give the power to become the sons of God. (John 1:12)

John 1:1-4 In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him; and without him was not any thing made that was made. In him was life; and the life was the light of men.

So many portions of Scripture attribute a divine, ever-existent quality to the Lord Jesus Christ, but Revelation 1:11 quotes the Lord as being "the Alpha and Omega".

Let us Worship Him for who He is, the Eternal Son Of God, Second Person of the Godhead, Who alone has power to wash away sins and give eternal life.

The closest belief in antiquity to what you are suggesting is, I believe, Nestorianism, which taught more or less what you pose: that Christ was less than 100% God when He was Incarnate, but was otherwise divine. Arianism - the denial that Christ had any divine nature whatsoever - is a related belief, but not quite the same as what you are asking about.

The root of the Nestorian controversy seems to have been not begun directly with a deliberation of the Godhead, but rather over what the appropriate description for the Virgin Mary should be. Nestorians and proto-Nestorians following Theodore of Mopsuestia (5th century) objected to referring to Mary (in Greek) as Theotokos - "God-bearer". They insisted that she should be called instead Christotokos - "Christ-bearer".

Nestorianism, as another answer has alluded, was condemned by the Third Ecumenical Council in 431. A chief defender against Nestorius was the bishop Cyril of Alexandria, who wrote a 5-volume book against the Nestorian doctrine. These volumes contain extensive exegeses of relevant Scriptures, refuting Nestorius' own interpretation of various "proof texts".

Yes, according to the doctrine of the Trinity, Jesus was always 100% God - including his time here on earth. In fact, the idea that he was not completely, 100% God was what led to the Arian controversy which precipitated the official forumlation Doctrine of the Trinity in the first place.

In fact, the original creed (before it was revised at the Council of Constantinople) included the line

But those who say: ... 'The Son of God is created,' or 'changeable,' or 'alterable'— they are condemned by the holy catholic and apostolic Church.

One of the options explored at the council of Nicea was that perhaps Jesus was ὁμοιούσιος ("Of a similar substance") as God (eg, 50% god), and this was explicitly rejected by the council. The final line of the creed makes it clear that this had been true for all points in time. So the doctrine does not support the idea that Jesus was at some point in tome 50% (or some other fraction) God.

The Chalcedonian Definition (AD 451) was a key christological statement of the early church, and is respected throughout trinitarianism. The first portion of the part relevant to this question emphasizes that Jesus is "truly God and truly man":

Following, then, the holy Fathers, we all unanimously teach that our Lord Jesus Christ is to us One and the same Son, the Self-same Perfect in Godhead, the Self-same Perfect in Manhood; truly God and truly Man.

But perhaps his divinity was reduced? The creed continues:

[Jesus is] acknowledged in Two Natures unconfusedly, unchangeably, indivisibly, inseparably; the difference of the Natures being in no way removed because of the Union, but rather the properties of each Nature being preserved and (both) concurring into One Person and One Hypostasis; not as though He were parted or divided into Two Persons, but One and the Self-same Son and Only-begotten God, Word, Lord, Jesus Christ.

So no, your understanding is decidedly not trinitarian – Jesus was simultaneously 100% God and 100% man while on earth; his divinity was not compromised.