Thinking is not what makes humans special according to the Christian faith. Christians usually refer to either Genesis 1:27 or 2:7.

1:27

“So God created man in his own image,
in the image of God he created him;
male and female he created them.”

2:7

“Then the LORD God formed a man [Hebrew "adam"] from the dust of the ground and breathed into his nostrils the breath of life, and the man became a living being.”

Because humanity bears God’s image, and because the “breath of life” was breathed into man, Christians believe humanity is special. The word breath, in Hebrew is actually also the same word as wind and spirit.

So it is inferred that the basis of human worth is in our reflection of God’s likeness and in the existence of God’s Spirit in us (which we can’t prove). At face value, this is an affront to the concept of human depravity… and therein lies the paradox, that humanity is both beautiful and depraved at the same time, and due to the current state of humanity, our worth is redeemed only in the restoration of our relationship with God.

If robots can have a relationship with God, then that’s another discussion.

Those two issues you raise are inconsequential to the existence of God, because God can exist without our comprehending God’s nature (relating to your question about omnipotence), and definitely our basis for believing in His existence.

I don’t even think we can subject human nature to logic, much less God’s nature.

And you cannot provide a reason for faith any more than you can prove miracles… and Jesus’ resurrection is the historical basis for Christianity.