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;
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.
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
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!”