They were chosen like everyone else in the bible so that they become a learning to the church.
Yes to show that a person can leave the faith and abandon God.
God is faithful, but we may not be.
2 Timothy 2:13
If we are faithless, (he remains faithful)— for he cannot deny himself.
1 Corinthians 10:13
No temptation has overtaken you that is not common to man. (God is faithful), and he will not let you be tempted beyond your ability, but with the temptation he will also provide the way of escape, that you may be able to endure it.
1 John 1:9
If we confess our sins, (he is faithful)and just to forgive us our sins and to cleanse us from all unrighteousness.
Matthew 10:33
33 but whoever denies me before men, I also will deny before my Father who is in heaven.
Matthew 26:14-16 ESV / 47
Then one of the twelve, whose name was Judas Iscariot, went to the chief priests and said, “What will you give me if I deliver him over to you?” And they paid him thirty pieces of silver. And from that moment he sought an opportunity to (betray) him.
Luke 22:54-62
Peter Denies Jesus
54 Then they seized him and led him away, bringing him into the high priest's house, and Peter was following at a distance. 55 And when they had kindled a fire in the middle of the courtyard and sat down together, Peter sat down among them. 56 Then a servant girl, seeing him as he sat in the light and looking closely at him, said, “This man also was with him.” 57 But he denied it, saying, “Woman, I do not know him.” 58 And a little later someone else saw him and said, “You also are one of them.” But Peter said, “Man, I am not.” 59 And after an interval of about an hour still another insisted, saying, “Certainly this man also was with him, for he too is a Galilean.” 60 But Peter said, “Man, I do not know what you are talking about.” And immediately, while he was still speaking, the rooster crowed. 61 And the Lord turned and looked at Peter. And Peter remembered the saying of the Lord, how he had said to him, “Before the rooster crows today, you will deny me three times.” 62 And he went out and wept bitterly.
Peter loved Jesus-
John 21:15-17
When they had finished breakfast, Jesus said to Simon Peter, “Simon, son of John, do you love me more than these?” He said to him, “Yes, Lord; you know that I love you.” He said to him, “Feed my lambs.” He said to him a second time, “Simon, son of John, do you love me?” He said to him, “Yes, Lord; you know that I love you.” He said to him, “Tend my sheep.” He said to him the third time, “Simon, son of John, do you love me?” Peter was grieved because he said to him the third time, “Do you love me?” and he said to him, “Lord, you know everything; you know that I love you.” Jesus said to him, “Feed my sheep.
Jesus then told Peter the kind of death he was to receive-
John 21:18-19
Truly, truly, I say to you, when you were young, you used to dress yourself and walk wherever you wanted, but when you are old, you will stretch out your hands, and another will dress you and carry you where you do not want to go.” (This he said to show by what kind of death he was to glorify God.) And after saying this he said to him, “Follow me.”