In what way(s) is God the Father a father to God the Son? And in what way(s) is God the Son a son?

(I'll interpret the use of "Trinity" and "Trinitarian" in this question broadly, as meaning non-unitarian (to include the binitarian view), since neither the question nor this answer involve any aspect of a third person.
But if the question does require the 3-in-1 doctrine, I doubt it will get a satisfying answer.)

In what sense is God the Father a father to God the Son?

Their relationship is the very definition of what the physical human relationship only symbolizes.

In what sense is God the Son a son of God the Father?

Human fathers and sons are physically and mentally very similar.

At least they are on the average. Yes, one might be a little taller, or slightly better at playing a piano, or …, but those are minor random variations. Humans are made in the image of God, but as physical beings they are only imperfect physical analogues of what God actually is.
And yes, humans do change with age (a father is physically and mentally superior to an infant, and a son physically and mentally superior to a feeble old man).
The point is that there is not necessarily anything that inherently makes one superior to the other.

The Father and Son are eternal and do not change or age, and they are very similar in every way, having the same form and possessing the same abilities. Each of them is "God":

• [Jesus] said also that God was his Father, making himself equal with God. — John 5:18
• I and my Father are one. — John 10:30
• he that hath seen me hath seen the Father — John 14:9

Both for imperfect father/son and for perfect Father/Son though, the fundamental attribute that distinguishes their relationship is an attitude of respect and deference on the part of the son.

If your answer is simply "the risen Jesus is subordinate to his Father and of the same nature", please explain how this makes him more of a son than the angels.

No created angel can ever be like the Father. Angels are spirit rather than flesh, but angels are not of the same nature as God; even ignoring their limited spiritual power, they simply do not have God's personality. ("For unto which of the angels said he at any time, Thou art my Son, this day have I begotten thee? And again, I will be to him a Father, and he shall be to me a Son?" — Hebrews 1:5)

But, just as a human son is of the same kind as his human father, the Son is of the same kind as the Father. The Son and Father have always shared the same power, ability, and personality, but the Son defers authority to the Father.

Of the relational aspects in your answer, which of these overlap with the concept of fatherhood and sonship within the design of nature and more specifically mankind?

All of them. The entire relationship between parents and children is a type of the relationship between the Father and the Son. This is not an accident; the family is God's deliberate creation.

From a Christian perspective, the entire purpose of life is to enter into that same relationship, preparing one's spiritual self to be reborn as an immortal spirit being, a younger sibling of the Son ("the firstborn among many brethren" — Romans 8:29), who as the eldest child holds a position deserving respect and deference.

The first thing to clarify is that Jesus’ sonship has nothing at all to do with biology. In using this language of Father and Son, the scriptures are drawing on 1st century human categories to (try to) describe what the connection is between God the Father and God the Son.

The “son-ship” of Jesus to God the father is primarily a role. It is drawing on the human categories of father and son as they existed in the 1st century and is primarily about rights of inheritance, and the position of representing the father. For example, in ancient times a father could send (especially) his firstborn son as a kind of family ambassador to represent his father in business dealings. In such a role, the son was understood to speak with the full authority of the father and could close deals without having to confirm with the father that he was in agreement.

So, Jesus holds this kind of role with regards to God. When Jesus speaks, the Father speaks.

There is also some sense in which the son, like the Holy Spirit - proceeds from the Father. He is distinct from the Father, yet is of one substance with him.

I don’t think anyone really understands what that means (Trinitarian theology has been wrinkling brains for two millennia now), but it is to be expected that an eternal, infinite God would be far more complicated than finite human brains can make sense of.

All we can do, is make assertions about the relationship, based on the evidence of scripture, in the ways Jesus interacted with God the Father, and the things his authorised Apostles (who spent several years with him) said was true about God.

Those assertions include that:

1. The Father is God
2. The Son is God
3. The Spirit is God
4. The Father is not the Son or the Spirit
5. The Son is not the Spirit
6. There is only one God

Somehow these six assertions are true, and we can made decisions about how we live assuming their truth, even if we don’t understand how they are true.

Yes, the Bible speaks of one and only one God because the Bible is quite explicit that there is only ONE God. The Father speaks of a Son because according to Him (and to scripture) this God had a Son an extension of Himself of HIS OWN NATURE just like all sons.

The "obvious" problem is many make "the Father" the ONLY reference to "God," when it is a term clearly employed throughout the New Testament to "distinguish" between the Father in His "God-ness," the Son in HIS same "God-ness," and the Spirit in His self-same "God-ness."

If "the Father" alone is God, there would be no need to distinguish Him as "the Father" and one would ONLY see the same continued metaphorical applications of that term. But you don't; God is NEVER ONCE identified as "the Father" in the Old Testament (other than in a collective and metaphorical sense).

God was called father (mainly and exclusively) by the Old Testament people because He is their creator. In the New Testament He is constantly so distinguished. so why bother making the distinction if there is no distinction to be made?

Since the Father of Christ is God, then Christ is God--He is the Son of God--who, and like all sons He bears the full ONTOLOGICAL NATURE of His Father. If this is not so, then the Bible is a lie, or at best an incredibly bad case of special pleading requiring that in this ONE instance in all of the universe a son actually does NOT bear the full nature of his own father.

Furthermore, the Son is subordinate to the Father. This is an intentional sonship and does not mean they are not equally God. There was a reason why Jesus Christ Himself often referred to Himself as "the Son of God" and as "the Son of Man." He's the "Son of God" on His Father's side and the "Son of Man" on His mother's side.

Overview
Let's begin by assuming these three statements are correct:

• Every father to ever exist has preceded the begetting of his child.
• Every son to ever exist must be begotten at a moment in time.
• A son inherits qualities and responsibilities from his father at a specific moment in time.

These may be displayed graphically on a time-line:

The first dotted line depicts a point in time at which the son comes into existence. The second shows the point at which qualities and responsibilities are inherited.¹

However, the OP improperly uses the title of "father" before there is a son. Yes, every father pre-exists every son, but until there is a first son, he who will be father, lacks that title:

It is at the same moment in time the first son is begotten the title of "father" may be used. Before then the proper title is "not a father" or "desires to be a father." Despite all intention and desire, he who is without a child is not a father.

Son of God
If we believe the Son of God was not eternal, then prior to His existence, God was without Son.²Therefore, the title of "Father" cannot be used with God, until the moment in time there is a Son. Just as the Son has a specific point at which He was begotten, God had a specific point in time at which He became Father:

Since God became something different, Father, when the Son came into existence, God must have been something less, "God," until He became Father. The problem with this is obvious.

On the other hand, if we understand the title of Father as one which may be used of God from all eternity past, then God must have had a Son from all eternity past:

The only way to recognize God the Father as always and eternally "Father" is is to recognize the Son is always and eternally "Son."

<sub>1. The two are illustrated taking place at different times. Both could take place at the same time.
2. The same is true if angels or some other being is created first. Before God creates the first which may be called son, God cannot be called "Father."</sub>

This may be an oversimplification, but here goes.

In Islam, good muslims insist that Allah is one God. To associate any other person with Him and claim this other person is also God is blasphemy to muslims. While you are not a muslim, I think you have this "one God" thing in common with muslims. While they do not address Allah as Father, the Judeo-Christian scriptures do.

As the hymn says, "Eternal Father, strong to save . . .." To say that only the Father existed for eternity but the Son did not is an impossibility. If God indeed is love (1 John 4:8), and has been from all eternity, then he MUST have had the Son to love from all eternity.

To say that at some point in time, the Father begot the Son is certainly not scriptural. Love cannot exist in a vaccuum. Love requires an object or a person to be loved--a person to receive that love. In other words, God cannot be love (again, 1 John 4:8) without an eternal Son to love.

As for how the Holy Spirit fits into the doctrine of the trinity, frankly I do not know. What I do know is that the Holy Spirit, who is called by the apostle John "the Advocate" and "the Spirit of prophecy who bears testimony to Jesus” (see John 14:16; 14:26; 15.26; 16:7; and Revelation 19:10) is fully God and is therefore eternal (see Acts 5:3-4).

As difficult as comprehending "three in one" may be, I submit that the Father--however you define His Fatherness--was never alone. Rather, He, the Son, and the Holy Spirit had a love fest going on from all eternity. God could not exist in any other way than as a Triune God: Father, Son, and Holy Spirit (see Matthew 28:19).

. . . . . . . 'in what sense' . . . . . ?

Note after question edit : The 'way' is the same as 'sense'. The words 'Father' and 'Son' mean something to them that believe in God. But the concept is not appreciated by those who do not perceive the Spirituality of the Deity.

Changing the word does not alter the matter of perception : the seeing or not seeing of God as he truly is.

The human 'senses' are not capable of either perceiving or understanding that which is eternal ; that which is of Spirit ; or that which is Divine.

None of the human 'senses' (which are created, not begotten of Spirit) are able to communicate with the invisible, the unhearable, the untouchable, and the unthinkable.

But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned. [1 Corinthians 2:14 KJV]

In other words : the things of God, who is Spirit, make no sense to the natural man. Or, according to the edited question : the way in which they operate is meaningless to the natural man.

This is why the natural heart and mind of man can neither appreciate, nor enter into, the knowledge of Deity ; and, most particularly, the relationship of (as you, yourself, precisely state) 'God, the Father' and 'God, the Son : words which the human mind cannot possibly process, there being 'One God'.

And who shall perceive these things if they are not born of God and are not, themselves, sons of God ?

For as many as are led by the Spirit of God, they are the sons of God. [Romans 8:14 KJV]

This is the very reason so many struggle with the concepts. That is to say either the 'sense' of them or the 'way' in which they can be understood and rested upon in faith.

As Jesus of Nazareth said 'Ye must be born again'.

Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God. [John 3:3 KJV]

And if not even the kingdom can be perceived, how much less the One who sits upon the throne ?

The knowledge of Deity requires repentance, that is to say 'another mind' (metanoia) than the carnal mind. And then faith is required, which is a gift of the Spirit, not a natural capacity.

Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost. [Acts 2:38 KJV]

... for by grace are ye saved through faith; and that not of yourselves: it is the gift of God: [Ephesians 2:8 KJV]

So "in no sense" is my answer.

Or, according to the edit, the 'way' in which the relationship between 'God, the Father' and 'God, the Son' is perceived cannot be perceived by nature but only by entering into that relationship oneself, by (Spirit-given) penitence and by (God-given) faith

There is no 'sense' which will aid you in your quest.

There is no 'way' for the natural man to tread upon. Strait is the gate and narrow is the way which leadeth unto life. And few there be which find it, said He who, himself, is that way, Matthew 7: 13,14.

These deeply spiritual, divine and eternal matters make no sense to those who are not the sons of God, born of God, by the work of the Spirit in repentance and faith, and gifted by the Spirit - once saved - to enter in fully, in understanding and spiritual apprehension, to the deep things of the eternal, unchanging God who made the heavens and the earth.

This knowledge is out of the reach of natural man : a sword turns every way, fiery in nature and wielded by angelic power.

So he drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life. [Genesis 3:24 KJV]

It keeps the way to the Tree of Life.

That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. [John 3:6 KJV]

Only that which is begotten of Holy Spirit can enter in and receive.

Natural man, Adam, is banished and is not only not permitted to reach out and grasp at Life, he is prevented from even attempting to do so.

I write these things as a lifelong Trinitarian (and Protestant) baptised at the age of five, in 1956, voluntarily, into the Church of Scotland (I remember the occasion vividly) and then baptised, again voluntarily, as an adult, into the Baptist Assembly of Scotland at the age of sixteen in 1967.