It would be most helpful if you would ask your questions or state your positions one at a time.
I will try to respond as you have posted but I do not do well with several things on one post.
#1. You asked...
Would you be kind enough to provide as example of unbelief ?
The first one I thought of is in the first book of the Bible, Adam and Ev
e. In the Garden of Eden, God gave Adam and Eve one command regarding the fruit of the tree of the knowledge of good and evil. However, their doubt and curiosity led them to question God’s word and ultimately disobey His command.
#2.You asked....
"What if those works are sinful ?"
Works done for God in the name of the Lord Jesus Christ can never be a sinfull act so your question does not apply.
#3.. You asked...
Will God turn a blind eye and judge unrighteously ?
God is holy and righteous and it is impossible for Him to do anything that is unrighteous. Now, as a believer YOU should know that and not have to ask.
#4. You asked....
Your outlook lacks the factors of rebirth from God's seed.
His seed cannot bear liars, thieves, or adulterers.
Your question lacks knowledge of how God works.
According to the Bible, the seed of God is the Word of God.
The Bible also says that God regards us as His seed.
This implies that if we are born again in the truth and grow in faith, we must resemble God and bear fruit.
Unless we learn God’s love and sacrifice correctly, we will not be able to fully perform our role and function as the seed of God which we are by FAITH spiritually in Jesus as the Christ.
#5.
What if he is reborn as a frog? What if he is reborn as a snail ? Will he still hunt small rodents ?
Your question tells me that you do not have any concept of what the Bible calls....."Born Again".
Born Again has nothing whatsoever to do with reincarnation or physical birth.
As Jesus talked with Nicodemus, He said, “‘I tell you the truth, no one can see the kingdom of God unless he is born again.’ ‘How can a man be born when he is old?’ Nicodemus asked. ‘Surely he cannot enter a second time into his mother’s womb to be born!’ Jesus answered, ‘I tell you the truth, no one can enter the kingdom of God unless he is born of water and the Spirit. Flesh gives birth to flesh, but the Spirit gives birth to spirit. You should not be surprised at my saying, “You must be born again”’” . "Born Again" is a SPIRITUAL condition in which the last sinner becomes a saved sinner by faith in Jesus as the Christ.
#6. You asked.
He sins because he is born of Adam's seed.
Correct. That my friend is called in theology, "Original Sin".
Original sin can be defined as “the moral corruption we possess as a consequence of Adam’s sin, resulting in a sinful disposition manifesting itself in habitually sinful behavior.” The doctrine of original sin focuses particularly on its effect on our internal nature and our standing before God. There are three main views that deal with that effect:
#7.....You asked
Those reborn of God's seed don't commit sin.
No Sir, that is not true.
Romans 3:23....."ALL have sinned and come short of the glory of God". ALL = Every human being!
#8. You asked....
Where does the sinner get confidence to think he has any relationship with God while he is lying, stealing, or committing adultery?
"For God so loved the world that He gave His only begotten Son that all who believe in Him will have ETERAL life"!
God's love overpowers and forgives the sinner of his sin.
#9.
It is possible to live without sinning.
NO!. Every human is affected by original sin and all have sinned and God does not approve of sin.
I will be glad to respond to your questions again BUT ONLY ONE AT A TIME.