What brought things into existence?
If existence is in some sense infinite, then the question of what created it becomes somewhat meaningless.
However, if there is a valid answer to that question, my answer is 'I don't know'. and nor do you actually. No doubt you would seek to assert your God as your answer, but as you seem totally incapable of giving any valid evidence for the existence of this God of yours, I have to come to the conclusion that you haven't any evidence and your God only becomes one of many possibilities. indeed, as others have noted, you shoot yourself in the foot by suggesting that everything that exists has a source and then being totally unable to respond to the logic that if your God exists then He must also logically have a source. Your pitiful response of Him being the ultimate source is simply special pleading and defies the logic that you have attempted to create.
What is responsible for the laws of physics and chemistry?
See previous response.
What is responsible for the incredible fine tuning of the forces and materials needed to bring life into existence?
It would only be incredible if there were not a vast amount of scenarios where it were possible and that we do not know. Also, it is pretty obvious that the only awareness of such a phenomenon would be exactly where life existed. (See Douglas Adams Puddle Analogy)
What facilitated the billions of beneficial mutations which brought us into existence?
Darwinian evolution is a perfectly valid response here, with plenty of evidence to support it.
What inspired the human race to seek reason and purpose beyond mere survival?
Although there are many different ideas on how our sense of reasoning and purpose developed, I would suggest that its origins lie in the complexities of our brains shaped by evolutionary processes to enable us better to survive, especially in group situations.
Discovering how things work does not exclude evidence of the God who made things work and who has made Himself known to us in the person of Jesus Christ.
Of course it doesn't, but this statement posits a God where there is no valid evidence that such a Being exists. As for Jesus, I take the view that there probably was an itinerant and charismatic preacher who formed the basis of the Gospels, but, as to Him being God or the son of God, no evidence whatever.