On what grounds was Arius refuted at the First Council of Nicaea?

There appears to be a misapprehension of what the Trinity actually teaches. So that to answer some of the questions here posed would be to implicity accept as true something never posited for belief.

For instance,

What proof does anyone have that Christ was the same and identical as the one He called "Father"

No trinitarian Christian believes that Christ is the Father, any more than light is the fire which gives it forth (to use a limited analogy).

The Father, the Son and the Holy Ghost are distinct Persons. The Father is not the Son. The Son is not the Spirit.

But they equally share one nature: being God.

Yes, Jesus was the Son of God although He never Himself said as much; He said He was "the Son of Man" only

This is just factually false.

It would suffice that the Holy Ghost said as much:

E.g.

Mark 1:1 (DRB) The beginning of the gospel of Jesus Christ, the Son of God.

John 20;31 But these are written, that you may believe that Jesus is the Christ, the Son of God: and that believing, you may have life in his name.

But Jesus said He had said He was the Son of God, and also testifies under solemn oath that He is:

Mark 14:61 But he held his peace, and answered nothing. Again the high priest asked him, and said to him: Art thou the Christ the Son of [God]? And Jesus said to him: I am. And you shall see the Son of man sitting on the right hand of the power of God, and coming with the clouds of heaven.

John 10:36 Do you say of him whom the Father hath sanctified and sent into the world: Thou blasphemest, because I said, I am the Son of God?

Note that His usage of 'the Son of Man' is the specific 'son of man' of Daniel 7 who recieves divine worship and is coming on the clouds of heaven, a famous perogative of the God Almighty alone. No one goes around saying they are a human being as their title. The title is in reference to a specific someone Daniel saw coming like God, yet looked as though He were a mere man. That is, Jesus Christ who took on "the form of a slave," being made in human likeness and yet formerly was in "the form of God." (Philippians 2:5-11 cf. v. 11 & Romans 14:11; Isaiah 45:23!).

Son of Man does not preclude that in addition to being human, Christ is God. For in Revelation we read, for example:

Revelation 1:13-18 And in the midst of the seven golden candlesticks, one like to [a] son of man, clothed with a garment down to the feet, and girt about the paps with a golden girdle. . . . And when I had seen him, I fell at his feet as dead. And he laid his right hand upon me, saying: Fear not. I am the First and the Last, and alive, and was dead, and behold I am living for ever and ever, and have the keys of death and of hell.

And 'Son of God' itself doesn't mean what non-trinitarians sometimes imply ('a favorite huamn son,' 'a creature') but the eternal Word or Son of God, Who has always been God.

Consider:

Revelation 2:18-23 And to the angel of the church of Thyatira write: These things saith the Son of God, who hath his eyes like to a flame of fire, and his feet like to fine brass. 19 I know thy works, and thy faith, and thy charity, and thy ministry, and thy patience, and thy last works which are more than the former. 20 But I have against thee a few things: because thou sufferest the woman Jezabel, who calleth herself a prophetess, to teach, and to seduce my servants, to commit fornication, and to eat of things sacrificed to idols. 21 And I gave her a time that she might do penance, and she will not repent of her fornication. 22 Behold, I will cast her into a bed: and they that commit adultery with her shall be in very great tribulation, except they do penance from their deeds. 23 And I will kill her children with death, and all the churches shall know that I am he that searcheth the reins and hearts, and I will give to every one of you according to your works.

This is the language of God:

Jeremiah 17:10 I am the [LORD] who search the heart and prove the [minds]: who give to every one according to his way, and according to the fruit of his devices.

Cf. Romans 2:6.

As to the issue of the Church declaring its Faith and doctrine authoritatively, this is simply the perogative of Christ's Church.

What does Christ say about those who will not heed the Church?

Matthew 18:15-18 15 But if thy brother shall offend against thee, go, and rebuke him between thee and him alone. If he shall hear thee, thou shalt gain thy brother. 16 And if he will not hear thee, take with thee one or two more: that in the mouth of two or three witnesses every word may stand. 17 And if he will not hear them: tell the church. And if he will not hear the church, let him be to thee as the heathen and publican. 18 Amen I say to you, whatsoever you shall bind upon earth, shall be bound also in heaven; and whatsoever you shall loose upon earth, shall be loosed also in heaven.

Christ gives them the authority in heaven and earth to "bind" others to adherence to that decree. A power of binding and loosing was not unknown (e.g. Matthew 22:2-3; see also the Jewish Encylcopedia entry for Binding and Loosing), although this decreeing something for heaven and earth denotes its universality, and its all-encompassing nature—and the indefectability of the body weilding said authority: since God cannot bind a lie in heaven, or even on earth.

This authority makes no sense in the 'nebulous' interpretation of the Church ( as being me and my Bible, and other people who read the Bible—'the Bible' didn't exist in the early centuries of the Church.) You can't 'take it to the Church' if that church is wherever you deem the place you belong theologically, and not the place assigned for you to become acclimatized to.

We see how this plays out very early in the Church's history, in Luke's history thereof (Acts).

Controversy/Heresy

Acts 15:1 "1 And some [men] coming down from Judea, taught the brethren: That except you be circumcised after the manner of Moses, you cannot be saved. "

Meeting of Bishops [Overseers] of the Church, i.e. a Council

15:6 "And the apostles and [elders] assembled to consider of this matter. "

Pope Settles Matter :]

15:8 "And when there had been much disputing, Peter, rising up, said to them: ..." 15:12 "And all the multitude held their peace..."

Council Document(s) Ratifying Findings

15:23-29

Infallibility of Outcome Declared

15:28 "For it seemed good to the Holy Ghost and to us..."

As for Constantine, he didn't have a say in the findings of the Council, but the bishops there (he only wanted unity in Christendom, which he legalized and thus felt responsible for); and no one at the Council denied that Jesus was divine, only how He was so, in relation to the Father.

The records of the 1st Ecumenical Council do not, I believe, include the discourse that preceded the condemnation against Arius and his followers.

We do know, however, that Athanasius of Alexandria was key in the condemnation. He had written extensively against Arianism in his Four Discourses Against the Arians. We can assume, I think, that many or all of the arguments from this document were brought to bear at the Council and/or that most of the Council members were familiar with them, even though the document itself seems to have been written after the council. I do not think it would be misstating things to say that the 1st Ecumenical Council was essentially an endors*m*nt by the whole Church of a previous deposition of Arius by the bishop Alexander within the See of Alexandria.

I provided a link to Athanasius' document above, which you see is quite comprehensive and detailed, with several sections devoted to exegesis of the New Testament Scriptures, contrasting what was understood to be the Orthodox interpretation with that of the Arians. An outline of the arguments against Arius is:

1. The Sonship of Christ is eternal

2. The Sonship of Christ is not like earthly Sonship

3. The personality of the Son is co-eternal with the Father

As you will see in the discourses, Athanasius was quite thorough. On each point, he considered the Arian's argument, and then considered and refuted the Arian's interpretation of each Scriptural "proof text" offered.