1. Given that there is God the Father, Jesus Christ the Son, and the Holy Spirit, is there a hierarchy in the Trinity?

The short answer is "yes" and "no." Consider the biblical evidence.

Yes - there is a hierarchy in relationship within the Trinity.

For this reason I bow my knees before the Father, from whom every family in heaven and on earth derives its name." (Eph 3:14-15)

God is recognized and honored as the One with Fatherhood.

"For God so loved the world, that He gave His only begotten Son, that whoever believes in Him shall not perish, but have eternal life." (John 3:16)

Jesus is recognized as being the Son with the appearance of coming after the Father and with unique characteristics: part man and fully God.

"For to us God revealed them through the Spirit; for the Spirit searches all things, even the depths of God. For who among men knows the thoughts of a man except the spirit of the man which is in him? Even so the thoughts of God no one knows except the Spirit of God." (1 Cor 2:10-11)

Thus the Holy Spirit knows and understands the thoughts and depths of God. And Jesus tells us in John 15:26, "When the Helper comes, whom I will send to you from the Father, that is the Spirit of truth who proceeds from the Father, He will testify about Me."

The Father sent the Son whose purpose was to reveal the Father. The Son sent the Spirit whose purpose was to reveal the Son.

No - no hierarchy exists in nature within the Trinity despite the different functional roles that They play.

Creation. Each personality of the Trinity created the world.

God the Father

"Of old You founded the earth, And the heavens are the work of Your hands." (Psalm 102:25)

Jesus Christ the Son

"For by Him all things were created, both in the heavens and on earth, visible and invisible, whether thrones or dominions or rulers or authorities-- all things have been created through Him and for Him." (Col 1:16)

Holy Spirit

"The earth was formless and void, and darkness was over the surface of the deep, and the Spirit of God was moving over the surface of the waters." (Gen 1:2)

Creation of man. Each personality of the Trinity created man.

God the Father

"Then the LORD God formed man of dust from the ground, and breathed into his nostrils the breath of life; and man became a living being." (Gen 2:7)

Jesus Christ the Son

"For by Him all things were created, both in the heavens and on earth, visible and invisible, whether thrones or dominions or rulers or authorities-- all things have been created through Him and for Him." (Col 1:16)

Holy Spirit

"The Spirit of God has made me, And the breath of the Almighty gives me life." (Job 33:4)

The Incarnation. The power of God, ministered through the Holy Spirit, resulted in the birth of Jesus Christ.

"The angel answered and said to her, 'The Holy Spirit will come upon you, and the power of the Most High will overshadow you; and for that reason the holy Child shall be called the Son of God." (Luke 1:35)

"Now the birth of Jesus Christ was as follows: when His mother Mary had been betrothed to Joseph, before they came together she was found to be with child by the Holy Spirit. And Joseph her husband, being a righteous man and not wanting to disgrace her, planned to send her away secretly. But when he had considered this, behold, an angel of the Lord appeared to him in a dream, saying, 'Joseph, son of David, do not be afraid to take Mary as your wife; for the Child who has been conceived in her is of the Holy Spirit." (Matt 1:18-20)

The Baptism of Our Lord. When Jesus Christ was baptized, the Spirit descended on Him and God the Father attested to His Son.

"But Jesus answering said to him, 'Permit it at this time; for in this way it is fitting for us to fulfill all righteousness.' Then he permitted Him. After being baptized, Jesus came up immediately from the water; and behold, the heavens were opened, and he saw the Spirit of God descending as a dove and lighting on Him, and behold, a voice out of the heavens said, 'This is My beloved Son, in whom I am well-pleased.'" (Matt 3:15-16)

Death of Christ. Each personality of the Trinity was involved with the death of Jesus Christ.

"how much more will the blood of Christ, who through the eternal Spirit offered Himself without blemish to God, cleanse your conscience from dead works to serve the living God?" (Heb 9:14)

The Resurrection of Christ. Each personality of the Trinity was involved with the resurrection.

God the Father

"For you first, God raised up His Servant and sent Him to bless you by turning every one of you from your wicked ways." (Acts 3:26)

"and to wait for His Son from heaven, whom He raised from the dead, that is Jesus, who rescues us from the wrath to come." (1 Thess 1:10)

Jesus Christ the Son

"Jesus answered them, 'Destroy this temple, and in three days I will raise it up.' The Jews then said, 'It took forty-six years to build this temple, and will You raise it up in three days?' But He was speaking of the temple of His body." (John 2:19-21)

"For this reason the Father loves Me, because I lay down My life so that I may take it again." (John 10:17)

Holy Spirit

"But if the Spirit of Him who raised Jesus from the dead dwells in you, He who raised Christ Jesus from the dead will also give life to your mortal bodies through His Spirit who dwells in you." (Rom 8:11)

The Provision of Our Redemption. Each personality of the Trinity is brought together in providing for man's eternal salvation. God the Father chose us for salvation, Jesus Christ the Son was the offering to God the Father through the power of the Holy Spirit, and the Holy Spirit set us apart.

"how much more will the blood of Christ, who through the eternal Spirit offered Himself without blemish to God, cleanse your conscience from dead works to serve the living God?" (Heb 9:14)

"Peter, an apostle of Jesus Christ, to those who reside as aliens, scattered throughout Pontus, Galatia, Cappadocia, Asia, and Bithynia, who are chosen according to the foreknowledge of God the Father, by the sanctifying work of the Spirit, that you may obey Jesus Christ and be sprinkled with His blood: May grace and peace be yours in fullest measure." (1 Pet 1:1-2)

Indwelling Believers. Both Jesus and God the Father dwell within Believers through the person of the Holy Spirit.

"Jesus answered and said to him, 'If anyone loves Me, he will keep My word; and My Father will love him, and We will come to him, and make Our abode with him.'" (John 14:23)

"And I will ask the Father, and He will give you another Helper, that He may be with you forever; that is the Spirit of truth, whom the world cannot receive, because it does not behold Him or know Him, but you know Him because He abides with you, and will be in you." (John 14:16-17)

The Great Commission. The name of each distinct personality of the Trinity is invoked at baptism; but note that they are declared in one name.

"And Jesus came up and spoke to them, saying, 'All authority has been given to Me in heaven and on earth. Go therefore and make disciples of all the nations, baptizing them in the name of the Father and the Son and the Holy Spirit, teaching them to observe all that I commanded you; and lo, I am with you always, even to the end of the age.'" (Matt 28:18-20)