How does the Catholic Church explain their belief that Jesus chose Peter to build upon?

Repeating the questions as I read them in the OP:

1. How does the Catholic Church explain their belief that Jesus picked faulty Peter to build upon?

2. Why does the Catholic Church adhere to this point of view, that Peter is the foundation that Jesus built his whole church on?

3. What is the reason that the CC adheres to this idea that Peter could be the foundation for the whole church of Jesus?

Rephrasing the answerable question to:

Why does the Catholic Church adhere to this point of view, that Peter is the foundation that Jesus built his whole church on, despite the record of his undeniable faults and sins?

Why does the Catholic Church teach that Peter is the foundation that Jesus built his whole church on?

Because Jesus chose him for this specific role. cf. e.g. CCC 551 - 553 and CCC 880 & 881.

CCC 551 From the beginning of his public life Jesus chose certain men, twelve in number, [...] As my Father appointed a kingdom for me, so do I appoint for you that you may eat and drink at my table in my kingdom, and sit on thrones judging the twelve tribes of Israel.

CCC 552 Simon Peter holds the first place in the college of the Twelve; Jesus entrusted a unique mission to him. [...]

CCC 553 Jesus entrusted a specific authority to Peter: [...] Jesus entrusted this authority to the Church through the ministry of the apostles and in particular through the ministry of Peter, the only one to whom he specifically entrusted the keys of the kingdom.

CCC 880 When Christ instituted the Twelve, "he constituted [them] in the form of a college or permanent assembly, at the head of which he placed Peter, chosen from among them." [...]

CCC 881 The Lord made Simon alone, whom he named Peter, the "rock" of his Church. He gave him the keys of his Church and instituted him shepherd of the whole flock. [...]

What are the reasons for Christ choosing Peter?

The LORD does as he pleases. The reasons why the LORD does things, he who does as he pleases, is in obedience to the will of the Father thereby rendering him - the Father - glory, and for love of his people, which again is God's will.

cf. John 17:6-7 (RSVCE) 6 “I have manifested thy name to the men whom thou gavest me out of the world; thine they were, and thou gavest them to me, and they have kept thy word. 7 Now they know that everything that thou hast given me is from thee;

And

Mark 3:13 (RSVCE) Jesus Appoints the Twelve 13 And he went up into the hills, and called to him those whom he desired; and they came to him.

And

John 15:16 (RSVCE) 16 You did not choose me, but I chose you and appointed you that you should go and bear fruit and that your fruit should abide; so that whatever you ask the Father in my name, he may give it to you.

Do Peter's faults and sins have any bearing on Jesus' choosing?

Not from this exchange between Peter and Jesus:

Luke 5:8-9 (RSVCE) 8 But when Simon Peter saw it, he fell down at Jesus’ knees, saying, “Depart from me, for I am a sinful man, O Lord.” 9 For he was astonished, and all that were with him, at the catch of fish which they had taken; 10 and so also were James and John, sons of Zeb′edee, who were partners with Simon. And Jesus said to Simon, “Do not be afraid; henceforth you will be catching men.”

Could Jesus have chosen anyone?

Yes! But no one chosen could have been faultless or sinless [the sinless one already had a vocation, that of being his Mother].

Romans 3:22-24 (RSVCE) 22 the righteousness of God through faith in Jesus Christ for all who believe. For there is no distinction; 23 since all have sinned and fall short of the glory of God, 24 they are justified by his grace as a gift, through the redemption which is in Christ Jesus,

It is good to end by noting that even at the very end of John's Gospel, it seems Peter is wondering about his election and commission, puzzling why it did not go to the disciple Jesus loved.

John 21:20-22 (RSVCE) Jesus and the Beloved Disciple 20 Peter turned and saw following them the disciple whom Jesus loved, who had lain close to his breast at the supper and had said, “Lord, who is it that is going to betray you?” 21 When Peter saw him, he said to Jesus, “Lord, what about this man?” 22 Jesus said to him, “If it is my will that he remain until I come, what is that to you? Follow me!”

The primary reason that the Church holds Peter as first Primate of the Church is that he was the first to confess who Jesus was:

Simon Peter holds the first place in the college of the Twelve;¹ Jesus entrusted a unique mission to him. Through a revelation from the Father, Peter had confessed: "You are the Christ, the Son of the living God." Our Lord then declared to him: "You are Peter, and on this rock I will build my Church, and the gates of Hades will not prevail against it." [Matthew 16:16–18]

(Catechism of the Catholic Church, paragraph 552)

¹Here there are a number of Scriptural references: Mk 3:16, 9:2; Lk 24:34; 1 Cor 15:5. These involve Peter being listed first among a number of apostles, or—in the last reference—seeing the Lord first after the Resurrection, before any others do.

Furthermore, the Church holds, Peter has a very specific authority over the Church and over Christian believers:

Jesus entrusted a specific authority to Peter: "“I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven." [Matthew 16:19] The "power of the keys" designates authority to govern the house of God, which is the Church. Jesus, the Good Shepherd, confirmed this mandate after his Resurrection: "Feed my sheep." [John 21:15–17] The power to "bind and loose" connotes the authority to absolve sins, to pronounce doctrinal judgments, and to make disciplinary decisions in the Church. Jesus entrusted this authority to the Church through the ministry of the apostles and in particular through the ministry of Peter, the only one to whom he specifically entrusted the keys of the kingdom.

All human beings have faults; but Jesus seems to have been one to not require faultlessness in order to merit a position in the Kingdom. The Church believes that despite Peter's quite apparent flaws, Jesus gave him authority over the Church because he was the first to declare Jesus as Lord.

Jesus was God and knew well the character and faults of Peter his disciple. The gospels give us a look at his impetuosity-asking Jesus 'if it is you, tell me to walk on the water', arrogance - wanting to call fire down from heaven, cowardice- denying the Lord, etc.

None of the twelve apostles were perfect. One doubted, one betrayed, one denied and all of them deserted him in the time when he most needed them. These men are far from perfect. Yet, in spite of themselves, God used them powerfully to change the known world in a single generation.

Since there are alternate explanations and meanings for Mathew 16:18

I don't know what alternative point of view you are talking about. But As far as I know, Jesus changed his name to Cephas (Yes, Cephas - as in Galatians 2:7-14, I Cor. 1:11-13, I Cor. 3:21, I Cor. 9:5 and I Cor. 15:5 meaning Rock not a pebble) has no other reasonable explanation. (Unlike James and John - the sons of thunder - , Cephas is not a name that denotes Peter's Character).

Peter seems unlikely as a foundation for the whole church .... Peter is not only unlikely from his character flaws that we see revealed in the gospels.

But the Lord said to Samuel, “Do not look on his appearance or on the height of his stature, because I have rejected him. For the Lord sees not as man sees: man looks on the outward appearance, but the Lord looks on the heart.” (1 Samuel 16:7)

But there could be many reasons why Jesus Chose Peter.

• Peter had a natural leadership skills (John 21:3)
• He was the first to act (John 20:6, John 21:7)
• He loved the Lord so much. (John 21:17)

...but any human person seems unlikely as a suitable foundation for the whole church of Jesus on the earth... why does the Catholic Church adhere to this point of view? What is the reason that the CC adheres to this idea that Peter could be the foundation for the whole church of Jesus?

Probably because Jesus himself said that he will built his Church on him and even went on to change his name as a proof of it. In the Bible, name of a person is changed by God when they are called to lead a significant role. (Like Abram-Abraham/Jacob-Israel) If Peter was just one of the twelve, this event is an anomaly.

Also all thru the New Testament, Peter's Primacy is shown in hidden symbols. Like for example,

In Gospels

• Peter's name appears first in every list of the twelve. Mathew even says he was the first (Mat 10:2)
• When evangelists mention the apostles, Peter is unusually singled out (Mark 1:36, Mark 16:7, Luke 9:32)
• Peter is approached by the temple authorities as a representative of the Group. (Mt 17:24-27 - Note this is the only place in the NT where Jesus refers himself in equal level with another human being. Jesus says '(Pay the tax for) me and for yourself')
• Peter often spoke to Jesus on behalf of the twelve. (Mk 8:29; Lk 12:41; Jn 6:66-69)
• At the Last supper, Even though Satan has asked to sift all of them, Jesus says he prayed for Peter specifically and asks him to strengthen others after regaining his own faith (Lk 22:31-32)
• John waited for peter to enter into the tomb. (John 20:6)
• Among the Apostles the resurrected Lord appeared first to Peter. (Luke 24:34; 1 Cor 15:5)
• Jesus made promises to Peter that he had not made to other Apostles. (Mt 16: 18-19; Rock/Keeper of Keys of Kingdom)
• Peter alone was given in charge to be the head shepherd (Jn 21:15-17)

In Acts

• Peter initiated and oversaw the replacement of Judas (Acts 1:15-26; Note that this happened before the decent of the Holy Spirit, i.e., he is acting on his own authority, given to him by Jesus)
• Peter delivered the inaugural sermon of Church History (Acts 2:14-36; First to preached and administered Baptism to the chosen people after resurrection)
• Peter was the first to preach and administered Baptism to the Gentiles (Acts 10:1-48)
• Peter performed the first healing after the resurrection. (Acts 3:1-10)
• Peter was the first to discipline (Acts 5: 1-11)
• Peter was the final authority in the first council of Jerusalem (Acts 15: 6-11)

In New testament

• Of all the twelve apostles Peter is mentioned more times than all other apostles put together. (Peter - 195 times; Other Apostles 130 times)

No other apostles has this unique mission of leading the Apostles/early Church as evident in the Bible.

[P.S: Also in your previous version of your question, you seem to confuse cornerstone with foundation rock. Corner stone is placed on Foundation rock to built upon. Both are different.]